· PDF fileu 5 100 13 5 11 2 1 = 500 32 =468 3. 1000 11 1000 8 (1000 3) + 1000 9 + 1000 21...

Number Systems 1

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In this Chapter following topics will be covered

Introduction

LCM and HCF

Units Digits and Remainders

Problems on Simple equations

This is the most important chapter in

quantitative aptitude for any competitive exams.

From this chapter, questions will come from

simplifications, divisibility rules, finding

remainders, finding unit digits (or last digits),

problems based on LCM and HCF…etc.

Natural numbers: Numbers which are used for

counting, it means all positive integers are natural

numbers. Examples are 1, 2, 3, and so on.

Based on divisibility there are two types of natural

numbers: Prime and Composite.

Prime Numbers: A number which doesn’t have

factors other than one.

Number 6 factors, are 1, 2 and 3. Factors for 8 are 1,

2 and 4 and factors for 7, only 1 and itself.

Therefore, 7 is a prime number.

Prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,

31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89,

and 97 so on. The only even number which is a

prime number is 2.

Note: For every prime number greater than 3,

generally will be in the form of 6 1n or 6 1n ,

where n is integer, but vice versa is not true, i.e.

numbers which are in the form 6 1n or 6 1n need

not be a prime.

To check given number is prime number or not:

Example: 181

Take a square root of the number, for 181, square

root lies between 13 and 14. Generally we round of

the square root to lower integer, i.e. 13.

Now, we note down all the prime numbers which are

less than 13 are 2, 3, 5, 7 and 11. If any of above

prime number divides given number i.e. 191, then

we say given number is not a prime.

If n is a prime number then, 2n – 1 is also a prime

number. 219

– 1 is prime since 19 is a prime number.

Composite Numbers: Any number other than prime

numbers called as composite numbers. Examples are

2, 4, 8, 10.so on.

Divisibility rules:

Divisibility by 2: Any number ends with even

number is divided by 2.

Examples are 298, 302, 1024 etc.

Divisibility by 3: If Sum of the digits in a number is

multiple of 3, then number is divided by 3. Example,

take a number 1236. Sum of the digits is 1 + 2 + 3 +

6 = 12, which is a multiple of 3 is so number is

divided by 3.

Check for 54749 and 987.

Divisibility by 4: A number formed with last two

digits in a number if it is divided by 4 then number

is divided by 4.

Example: 18548. Check if 48 is divided by 4, then

18548 is divided by 4.

Divisibility by 5: If a number ends with zero or five

then number is divided by 5. (eg.15, 20, 1255...).

Divisibility by 6: If a number is divided by both 2

and 3, then number is divided by 6.

Divisibility by 7: If the difference between the

number of tens in the number and twice of the unit

digit is 0 or divisible by 7, then number is divided by

7.

Take a number 987, double the unit digit is7 2 =14difference is98 14 84 , check if 84 is divided by

7. So number 987 is divided by 7.

Divisibility by 8: A number formed with last three

digits in a number if it is divided by 8 then number

is divided by 8.

The number 4816 is divisible by 8 because 816 is

divided by 8.

Divisibility by 9: A number divisible by 9, if sum of

the digits of a number is multiple of 9. Take an

example 60813, sum of the digits is 6 + 0 + 8 + 1 + 3

=18, is a multiple of 9.

Divisibility by 10: A number should end with 0.

Divisibility by 11: If sum of the alternate digits is

same or differ by multiples of 11. Take 1430,

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Number Systems 2


1 3 4 and sum of the other digits is also 4 (= 4 +

0) so number is divided by 11.

Take 587543, sums are 5 7 4 8 5 3 16

equal here so number is divided by 11. Check for

398849 and 5477319.

Divisibility by 19: If the addition of the number of

tens in the number and twice of the unit digit is

divisible by 19, then number is divided by 19. Take

684, 68 + 24 =76 is divisible by 19. If we have a

big number we continue above process step by step.

Whole numbers: Natural numbers and zero called

as whole numbers which are non-negative integers.

Real numbers: A real number is a value that

represents any quantity along a number line (number

line represents numbers from - to + ). Because

they lie on a number line, their size can be

compared. You can say one is greater or less than

another, and do arithmetic with them.

Rational numbers: A rational number is one that

can be represented as the ratio of two integers. For

example all the numbers below are rational:

1 3, ...

2 5

Irrational numbers: An irrational number cannot

be written as the ratio of two integers. Examples,

Pi = 3.14, e = 2.718, 2 =1.4142…

Simplify the following:

1. 89 + 19 + 39 + 97 + 78 + 101=?

2. 87 95 89 98 99 ?

3. 989 + 992 – 997 + 991 + 979 =?

Solutions:

1. 90 – 1 + 20 – 1 + 40 – 1 + 100 – 3 + 80 – 2

+ 100 +1 = 90 + 20 + 40 + 100 + 80 + 100 –

7 = 423

2. 90 3 90 5 90 1 + 90 + 8 + 90 + 9

= 468 Or 5 100 13 5 11 2 1

= 50032 =468

3. 1000 11 1000 8 (1000 3)

+1000 9 +1000 21 4000 1000 46 = 2954.

Example 1: 7845 + 3479 + 6432 + 5894 =?

A 24655 B 23650 C 23655 D 24660

Solution: Start with unit digits

5 9 2 4 20 0 carry 2

So, options A and C are eliminated.

4 7 3 9 carry 2 = 25

5 carry 2

Last 2 digits of the answer will have 50, option B is

correct. If one of the options is “none of these” then

we need to do addition till the last step.

Example 2: 9548 + 7314 = 8362 +?

A 8230 B 8410 C 8500 D 8600

Solution: Consider unit digits on both the sides.

8 + 4 = 12 = 2 +? so last digit of the answer should

be 0. Going to next step, i.e. adding the tens digits,

Then, 4 + 1 + carry 1 from previous step = 6 = 6 +?

So tens digit of the answer should be 0. Options A

and B are eliminated. Add the digits of the hundreds

place, 5 + 3 = 8 = 3 +?. 5 should be there. So option

C is correct.

Practice Set – 1:

1. What should be the maximum value of B in the

following equation, 5A9 – 7B2 + 9C6 = 823 A

6 B 5 C 7 D 9

2. When 335 is added to 5A7, the result is 8B2

which is divisible by 3. What is largest possible

value of A? A 8 B 2 C 1 D 4

3. If A381 is divisible by 11, find the value of

smallest number A? A 1 B 2 C 5 D 7

4. If x and y are the integers such that (3x + 7y) is a

multiple of 11, which of the following will also

be divisible by 11?

A 4x + 6y B x + y C 9x + 4y D 4x – 9y

5. What least number should be subtracted from

28038, so that the remaining will be divisible by

15? A 8 B 3 C 5 D 6

6. If a boy multiplied 36 by a number and obtains

8674 as his answer which is wrong. What is

correct answer? A 8672 B 8676 C 8678 D 8680

7. In four consecutive prime numbers, there are in

ascending order, product of first three is 385,


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Number Systems 3


and that of last three is 1001. The largest given

prime number is? A 17 B 19 C 11 D 13

8. Product of three consecutive numbers is whose

sum is 15? A 120 B 150 C 125 D 210

Suppose 33 divided by 5, 3 is the remainder. So we

write as 33 = 56 + 3.

In a division sum, dividend (N) = divisor (d)

quotient (q) + remainder (r).

Number = dq + r.

Example 3: If a number is divided by 4 gives

remainder 3, find the remainder if twice of that

number divided by 4? (S.S.C 2012)

Solution: Let number is n.

n = 4q + 3. 2n/ 4 = (24q + 23) = (8q + 6)/ 4.

Remainder is 2. If square of that number is divided

by 4, n2 = (4q + 3)

2 = 16q

2 + 24q + 9, remainder is 1

(since 9/4 gives remainder 1).

Example 4: Two numbers 19 and 14 are divided by

a number n leaves the same remainder. Find the

value of n? A 5 B 6 C 4 D 7

Solution: This kind of problems you can check with

options. Take the difference between 19 and 14, i.e.

19 – 14 = 5. It means if 14 and 19 divide by 5 or its

factors, leaves the same remainders.

9. In a division sum, the divisor is 3 times the

quotient and 6 times the remainder, if the

remainder is 2, then dividend? A 36 B 28 C 50

D 48

10. A number is divided by 72 gives 12 remainder

as, find the remainder if the same number is

divided by 9? A 5 B 6 C 3 D 4

11. If a number is divided by a divisor remainder is

24, if twice of the number is divided by same

divisor remainder will be 11. Find the divisor?

A 25 B 31 C 37 D 32

12. Two numbers 79 and 54 are divided by a

number n leaves the same remainder. Find the

value of n? A 10 B 5 C 25 D both B and C

13. Two numbers 4794 and 3378 are divided by a

number n leaves the same remainder. Find the

value of n? A 472 B 365 C 452 D none

Let’s take a 2 digit number, 43, reverse the digits

and subtract the number from original number.

43 – 34 = 9 (= 91).

Take one more example 72, 72 – 27 = 45 (= 9 5).

Example 5: Difference between a two digit number

and number obtained by reversing the digits of

actual number is always divisible by? (S.S.C 2013)

A 11 B 10 C 9 D None

Solution: Assume number is XY number obtained by

reversing the digits is YX.

So (10X + Y) – (10Y + X) = 9 (X – Y).

So difference always equals to 9 multiple.

14. Sum of the digits of a two digit number is 10,

while when the digits are reversed, number

decreases by 54. Find the changed number?

(S.S.C 2013) A 28 B 19 C 82 D 46

15. Raju had to do a multiplication, instead of taking

34 as one of the multipliers, he took 43. As a

result, the product went up by 540. What is

correct answer? A 2020 B 2040 C 1860 D None

16. How many two digit numbers are there to give a

perfect square number when they added to

number formed by reversing the digits? A 4 B 8

C 12 D 10

17. The difference between 791 and its reverse is

divided by 99, and then quotient is?

A 0 B 4 C 3 D 6

Important observation is whenever, if take the

difference between a number and number

obtained by reversing the digits of original

number is always 9 multiple.

Difference between a 3 digit number and

number obtained by reversing the digits of

actual number is, like 783 – 387 = 396 (= 99 x

4).

Let take a 3 digit number, XYZ, value of it is

100X + 10Y + Z. Number obtained by

reversing the digits is ZYX, value is 100Z +

10Y + X.

Difference is, (100X + 10Y + Z) – (100Z +

10Y + X) = 99 (X – Z) = 99 multiple.


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Number Systems 4


Squares:

Try to remember square of numbers from

1to 20. Procedure to find square of numbers ending

with 5, take some examples 15, 25, 35, 75, 95,

375…etc.

Example 6: Find the value of275 ?

Solution:

Step1: The last 2 digits of the answer is, 25 = 25.

Step2: Remaining part of the answer is, take 7 and

multiply it with next number i.e. 8, so it’s 78 = 56.

Join the two parts 25 and 56. 275 5625 .

It works for any number ending with 5. 235 1225 (Since 34 = 12) 295 9025 (Since 910 =90)

2195 38025 (Since 1920 = 380) 2 236 (35 1) 1225 1 2 35 1 = 1296

2 264 (65 1) 4225 1 2 65 1

2 289 (90 1) 8100 1 2 90 1

2 2 2 272 (70 2) ,61 (60 1)

2 287 (90 3) Or

2(85 2) .

Example 7: Find the squares of

A. 104

B. 109

C. 113

D. 98

Solutions:

A. Take 100 as base, since 104 is close to 100.

Difference is 104 – 100 = 4.The last two digits

of the answer is24 16 .Remaining part of

the answer is = 104 + 4 (difference) = 108 so

final answer is 10816.

B. Difference is 109 – 100 = 9.The last two digits

of the answer is 29 81 =109 + 9 = 118 so

final answer is 11881.

C. 113 – 100 = 13213 169 . The last two

digits of the answer is 69 and 1 goes as carry

for the next step. Remaining part is = 113 + 13

= 126 + carry (i.e.1) =127, so final answer is

=12769.

D. Difference is 98 – 100 = – 2. The last digit in

answer is – 22 = 4. = 98 + (–2) = 96 so answer

is 9604.

Square root of a perfect square number

Perfect square number ends with digits 0, 1, 4, 9, 6,

and 5 only. (12 = 1, 2

2 = 4, 3

2 = 9, 4

2 = 16, 5

2 = 25,

62 = 36, 7

2 = 49, 8

2 = 64, 9

2 = 81, 10

2 = 100)

Example 8: 2209 ?

A 43 B 47 C 57 D 53

Solution: Divide 2209 into two parts as 22 and

09. Number ending with 9 so last digit of the answer

is may be 3 or 7 (since23 09 2, 7 49 )

Take remaining part 22 which is 2 24 22 5 so we

can take 4. From above we can say answer is 43 or

47, but 2209 is close to 452 = 2025 so final answer is

47 (2209 > 2025(=452)

).

Get the answers:

A. Find the 7056 =?

B. Find the 4489 =?

C. Find the 2809 5929 ?

D. 17689 1849

?3249 1089

Square root of a non-perfect square number

Example 9:

1. 39 =?

2. 500 =?

3. 3125 =?

4. 93500 ?

5. 38472148 =?

Solutions:

1. We know 26 36 and

27 49 so answer

must be in the range of 6 to 7. Now start with

guessing. Let’s start with 6 divide 39 by 6.

396.5

6 and take averageof 6 and 6.5.


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Number Systems 5


6.5 66.25

2

. So 39 is 6.25 (Very

close approximate value).

2. We know that, 220 400 and

225 = 625 so

initial guess is 20.

500 20 2525 22.5

20 2

.

3. 2 255 3025,60 3600 , so expected

answer will be in the range of 55 to 60 and

that too close to 55, therefore start with

guessing 55.

3125 56.8181 5556.8181

55 2

55.90

4. Divide this into three parts as 9, 35 and 00.

For 9 take 3, for 35 take 0 and for 00 take 0

therefore our initial guess is 300. Divide

93500 by 300.

93500 311.666 300311.666

300 2

305.833

5. Divide this number into two digit groups as

48, 21, 47 and 38. For 38 take 6, for 47 take 0,

for 21 take 0, for 48 take 0 therefore divide the

given number by 6000.

384721486412.014

6000

6412.014 6000

2

= 6206.0123

Get the answers:

1. 270 ?

2. 380 132 ?

3. 1150 2864 ?

Example: 2 241 40 81 = (41 + 40) (41 40).

18. Difference between squares of two consecutive

numbers is 71, find the numbers?

A 35 and 34 B 36 and 37 C 36 and 35 D None

Example 10: Difference between the squares of two

consecutive odd integers is always divisible by

(MAT 2003) A 3 B 6 C 7 D 8

Solution: Take 5 and 7, 49 – 25 = 24.

Take 3 and 5, 25 – 9 = 16. So 8 is the answer.

Or 2x is even number, 2x + 1 is odd number, next

odd number will be 2x + 3. =2 2(2 3) (2 1)x x =

12x + 9 – 4x – 1= 8 (x 1)

Difference is multiple of 8.

Cubes

Try to remember cubes of numbers only from 1

to 9.

1³ = 1

2³ = 8

3³ = 27

4³ = 64

5³ = 125

6³ = 216

7³ = 343

8³ = 512

9³ = 729

10³ = 1000

One interesting property, please note the last digits

(unit digit for each numbers cube).

1³= 1 => last digit is 1






7³=343 => last digit is 3



10³= 1000 => last digit is 0

These are very useful in finding cube root of

a perfect cube number. If last or unit digit of the

perfect cube number is 0, then last digit of the cube

root is 0, if last digit of the perfect cube number is 2,

the last digit of the cube root is 8. (83

= 512, cube

ends with 2)

19. The least number which should be multiplied to

243 to get a perfect cube is (S.S.C 2012) A 2 B 3

C 6 D 9

Difference between squares of two

consecutive numbers always equals to sum

of the numbers.


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Number Systems 6


Key

Set -1: 1 C 2 D 3 D 4 D 5 B 6 B 7 D 8 A 9 C 10 C

11 C 12 D 13 A 14 C 15 B 16 B 17 D 18 C19 B

Finding cuberoot of a perfect cube number.

Example 11:

A. 3 9261 =? A 21 B 31 C 41 D 29

B. 3 12167 =? A 23 B 33 C 27 D 29

C. 3 51653 =? A 37 B 47 C 39 D None

D. 3 912673 =?

Solutions:

A. Divide the number into two parts first part

should be last three digits of the number, as

9 | 261 . Take the first part 261, last digit is 1,

so last digit of the answer should contain 1.

For remaining part of the answer, take second

part i.e. 9 note the highest cube less than

remaining number i.e. 9 here, 9. i.e. 2³ = 8 < 9.

So answer is 21. This method works out only

for perfect cube number. By this method, for 3 11061 will get same answer as 21 which is

wrong. If we have “None” as one of the options

we can’t use this method, we need to cross

check.

B. Divide the number into two parts as 12|167.

Take 167, last digit is 7, so last digit of the

answer must be 3 (since 3³= 27). 2³ = 8 < 12

therefore final answer is 23.

C. By above method, 51|653, take 653, so last

digit of the answer is 7 (since 7³ = 343). Since

3³= 27 < 51 therefore final answer is 37 which

is wrong. 373 = 50653 not 51653.

D. Solution: 912|673, for 673 its 7. For 912 its 9

(since9³ = 729 < 912) therefore answer is 97.

Get the answers:

A. 3 19683 =?

B. 3 39304 =?

C. 3 379507 103823 ?

D. 3 3

3 3

238328 778688?

24389 4913

LCM and HCF Let us write multiples of numbers 6 and 8.

For 6 6,12,18,24,30,36,42,48,54,...etc and

for 8 8,16,24,32,40,48,56,64,72....etc

There are some multiples which are

common for both i.e. 6 and 8. Common multiples for

6 and 8 are 24, 48, 72, 96, 120 or we can also write

them as 24, 224, 324, 4 24…etc. So common

multiples are in the form of k 24, where

1,2,3...k etc .

Here, 24 is the first (least) common multiple.

L.C.M means least common multiple, a common

multiple which occurs first, i.e. 24. Common

multiples are in the form of kLCM.

We can say L.C.M of two or more numbers is the

least number which is divisible by each of these

numbers (remainder is 0).

Write factors for 6 and 8.

61, 2, 3, 6 and for 81, 2, 4, 8.

Common factors are 1 and 2. Highest common

factor is 2.This is called as G.C.D or H.C.F of 6 and

8. H.C.F or G.C.D of two or more numbers is

highest factor or largest divisor of those numbers.

Example 12: Find H.C.F and L.C.M of 8, 12?

Solution: Take the ratio of the two numbers.

Simplify them to as minimum as possible. =

8 2

12 3 , from here we cannot simplify anymore.

To get H.C.F

= 8 12

42 3

or numerator

(numerator

denominator

denominatoror ).

L.C.M is 83 or 122 = 24(Cross multiplication)

Find H.C.F and L.C.M of following

1. 24 and 36?

2. 24 and 64?

3. 54 and 36?

Answers:

1. 24 12 4 2

36 18 6 3

H.C.F is24 36

122 3

or and L.C.M is 24 3 =

72.


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Number Systems 7


2. 24 12 6 3

64 32 16 8 ,HCF

248

3 , and

LCM is 248=192

3. 36 18 6 2

54 27 9 3 , HCF

3618

2 and

LCM is 542 = 108

Find H.C.F and L.C.M for following questions.

54 and 72?

36 and 42?

48 and 36?

52 and 72?

255 and 195?

Example 13: 1. Find L.C.M and H.C.F of 24, 36 and 42?

2. L.C.M and H.C.F of 150, 210 and 300?

Solution:

1. Method 1: we write above numbers in terms of

prime factors.324 2 3 ,

2 236 2 3 and

42 2 3 7 . To get H.C.F, write common

prime factors for all the three numbers, i.e. =

2 3 = 6. To get L.C.M, note all the prime

factors that appear at least once in any of the

numbers: 2, 3 and 7. Take their highest powers

in above numbers, = 3 22 3 7 = 504.

Method 2: Take any two at a time, 24/36 = 2/3.

H.C.F = 24/2 = 36/3 = 12. Now, 12/42 = 2/7,

H.C.F = 12/2 = 42/7 = 6. L.C.M = 243 = 362 = 72. Now, 72/42 = 12/7. L.C.M is 727 =

4212 = 504.

2. 150/210 = 5/7. H.C.F is 150/5 = 30 and LCM is

1507 = 1050. 30/300 = 1/10, H.C.F is 30.

1050/300 = 7/2, L.C.M is 10502 = 2100.

Example 14: The H.C.F and L.C.M of two numbers

are 12 and 144 respectively. Find the other number if

one of them is 36?

Solution: 12 144 = 36b, b = (12 144)/ 36 = 48.

Example 15: Find the smallest number which when

divided by 5 or 6, leaves a remainder of 2 in each

case and the number should be greater than 6 and 5?

Solution: Here, the L.C.M of 2 numbers is 5 6 = 30.

Hence the required number is 30 + 2 = 32. (Here,

same remainder in each case i.e.2)

Next numbers which satisfies above condition are 302 + 2 = 62, 30 3 + 2 = 92, 30 4 + 2 = 122

…etc. It means they are in the form of

L.C.M k + remainder, where k is 0, 1, 2, 3…etc.

Smallest 3 digit number which satisfies above

condition is 122.

Largest 3 digit number which satisfies above

condition is, first divide 999 by 30 (LCM) remainder

is 10, so number divided by 5 and 6 leaves

remainder zero is 990, and number is 990 + 2 = 992.

Smallest 4 digit number can be

990 30 1020 2 1022 .

To get largest 4 digit number, we need to divide

9999 by 30. If you divide 9999 by 30, 10 is the

remainder, so answer is 9990 + 2 = 9992.

Example 16: Find the smallest number which when

divided by 5 or 8, gives reminders of 3 and 6

respectively?

Solution: Here remainders are different. But the

difference between divisors and their remainders is

same in each case, 5 3 2 8 6 .

So answer for above problem is 40 – 2 = 38.

a and b are two numbers and relationship

between H.C.F and L.C.F of those two

numbers is,

H.C.FL.C.M = ab, i.e. product of

H.C.F and L.C.M of any two numbers

equals to product of those two numbers.

Any number which when divided by

2 or more numbers leaving same remainder

r in each case in the form of (LCM of those

numbers) k + r, where k is 0,1,2,3,…etc.

Any number which when divided

by a and b, gives remainders of c and d,

where a – c = b – d = constant difference

= (say) r, will be in the form of LCM of

those numbers) k – r, where k = 1, 2,

3…etc. (same approach for when more

than 2 numbers involved.)


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Number Systems 8


Practice set – 2

1. L.C.M of two numbers is 2079 and their H.C.F

is 27. If one of the numbers is 189, the other

number is (S.S.C 2013) A 189 B 216 C 297 D

584

2. The greatest 4 digit number exactly divisible by

10, 15 and 20 is? (S.S.C 2013)A 9990 B 9960 C

9980 D 9995

3. Find the largest 3 digit number which when

divided by 12 or 8, leaves a remainder 3 in each

case? A 987 B 982 C 984 D 997

4. The least multiple of 13 which when divided by

4, 5, 6, 7 leaves remainder 3 in each case is?

(S.S.C 2012) A 3780 B 3783 C 2520 D 2522

5. Find the largest 4 digit number which when

divided by 5 or 8, gives reminders of 3 and 6

respectively? A 9958 B 9996 C 9978 D 9998

6. Two alarm clocks ring their alarm at regular

intervals of 50 seconds and 48 seconds

respectively. When they ring together, if they

ring at 8.00 AM for the first time? A 8.15 am B

8.20 am C 8.25 am D 8.30 am

7. Two Traffic signals changes after 36 and 42

seconds respectively. If the lights are switched at

9.00AM at what time will they change

simultaneously? A 9.04.12 AM B 9.04.16 AM

C 9.06.12 AM D 9.04.20 AM

8. A red light flashes five times in two minutes and

a green light flashes three times in one minute at

regular intervals of time. If both the lights flash

at the same time, how many times flash together

in each hour? A 31 B 29 C 30 D 28

9. Two persons run in circular track takes 48 and

36 seconds to complete one full cycle. When

they meet for the first time on the track at

starting point if they are running in

A. Same direction B. Opposite direction

10. Find the area of smallest square which can be

formed with dimensions 75 cm? A 125 B 35 C

1225 D None

11. If the students of 9th class are arranged in rows

of 6, 8, 12 or 16, no student is left behind. Then

the possible number of students in the class is?

(S.S.C 2013) A 60 B 72 C 80 D 96

12. There is a number X which when divided by 3,

5, 7 and 9 leaves remainder of 1, but leaves a

remainder of 7 when divided by 8. Find the least

such number possible?

(A) 316 (B) 946 (C) 631 (D) 1261

13. To celebrate their victory in the World Cup, the

Germans distributed sweets among themselves.

If the sweets were distributed equally among the

11 players, 2 sweets were left. If the sweets were

distributed equally among the 11 players, 3

extras and 1 coach, then 6 sweets were left.

What could be the number of sweets in the box?

(A) 336 (B) 170 (C) 156 (D) 160

14. If a person makes a row of toys of 20 each, there

would be 15 toys left. If they made to stand in

rows of 25 each, there would be 20 toys left, if

they made to stand in rows of 38 each, there

would be 33 toys left and if they are made to

stand in rows of 40 each, there would be 35 toys

left. What is the minimum number of toys the

person have?

(A) 3805 (B) 3795 (C) 3185 (D)4285

Example 17: Two numbers are in the ratio 8: 5 their

HCF is 3, numbers are?

Solution: If you take ratio between any 2 numbers,

HCF will get cancelled. Suppose take 16 and 18. =

16 2 8 8

18 2 9 9

(Here HCF is 2 got cancelled).

For above problem go in reverse way, numbers are 83 = 24 and 53 = 15.

Example 18: Find the largest number with which

when 38 and 51 are divided gives remainders of 2

and 3 respectively?

Solution: Answer is HCF of (38 – 2) and (51 – 3)

i.e.12. (38 = 12 3 + 2 and 51 = 12 4 + 3).

Example 19: Find the largest number with which if

we divide 1000, 700 and 250, remainders are same?

The largest number with which when

a and b divided, gives remainders of c and d

will be the HCF of (a – c) and (b – d).


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Number Systems 9


Solution: 1000 – 700 = 300, 700 – 250 = 450

HCF of 300 and 450 is 150.

Example 19: Find LCM and HCF of2

3,

4

5?

Solution: LCM of numerators 2, 4 is 4.

LCM of denominators 3, 5 is 15.

HCF of numerators 2, 4 is 2.

HCF of denominators 3, 5 is 1.

LCM of fractions = 4

1

HCF Of fractions = HCF of numerators

LCM of denominators=

2

15

15. Two equilateral triangles have the sides of

lengths 102 and 51 respectively.

A. The greatest length of tape that can measure

both of them exactly is.

B. How many such equal parts can be

measured?

16. A farmer wants to plant 44 apple, 88 papayas

and 66 banana trees in equal rows. The number

of rows (minimum) that are required are? A 11

B 18 C 9 D 22

17. There are 315 and 420 students. What is the

minimum number of rooms required such that

all the rooms have equal number of students?

A 105 B 35 C 7 D None

18. The greatest number that will divide 19, 35 and

59 to leave the same remainder in each case is:

(S.S.C 2012) A 6 B 7 C 8 D 9

19. HCF of

2 4,

3 5 and

6

7 is? (S.S.C 2012) A

1

105 B

24

105 C

48

105 D

2

105

20. Three gold coins of weight 780gm, 840gm and

960gm are cut into small pieces, all of which

have the equal weight. Each piece must be heavy

as possible. If one such piece is shared by two

persons, then how many persons are needed to

give all the pieces of gold coins?

(A) 43 (B) 72 (C) 86 (D) 100

21. Find the numbers which when divided by 7

gives a remainder of 3 and when divided by 5

gives the remainder of 2.

A. Least number

B. Least 3 digit number

C. Largest 3 digit number

D. Least 4 digit number

E. Largest four digit number

Unit’s digits (Last digits)

Find the unit’s digit in the following expressions.

83 34 44 91 13 37 19

83 31 41 99 37 65

72 99 75 23 48 31 333 456

22. Unit digit of 3 38537 1256? (S.S.C CGL

2013) A 4 B 2 C 6 D 8

23. If the last digit in the product is 53 457 235n is

zero

A. What is the maximum possible value n may

take?

B. How many such possible values for n?

Find the number of zero’s in the expressions

15 33 42 8

2 5 2 6 42 5 4 15 8

1! 2! 3! 4! 5!1 2 3 4 5

24. Find the number of zero’s in the product 1, 3, 5,

7…99, 101 and 128? A 62 B 12 C 26 D 7

Largest number with which a, b and c

are divided gives same remainders will be HCF

of (a – b) and (b – c) or difference between any

two.

LCM of fractions = LCM of numerators

HCF of denominators

HCF of fractions = HCF of numerators

LCM of denominators


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Number Systems 10


Find the unit’s digits in the below examples.

199 2537 9 and 199 2537 9

199 253 199 1987 9 2 3

123456789123456789 ?

1554 4551(997) (799) ?

Remainders

Find the remainders in following problems

If 1415 divide by 8

If 135243362 by 12

76 78 79

80

1256 1258 1259

1260

Find remainders for following questions.

13562

9 And

13662

9?

153246

13 And

153336

16?

25. The remainder when 1737

+2937

is divided by 23,

is (S.S.C CGL 2012) A 17 B 29 C 0 D 1

26. The remainder when 6767 67 is divided by 68?

(S.S.C CGL 2001) A 67 B 1 C 0 D 66

Problems on Simple Equations

Example 19: Find the following

A. The product of two numbers is 120 and sum of

their squares is 289, find the difference of the

numbers?

Solution: Let numbers are x and y. xy = 120.

x2 + y

2 = 289. (x – y)

2 = 289 – 2120 = 49.

x – y = 7.

B. The sum of squares of two numbers is 80 and

the square of their difference is 36. The product

of the two numbers is?

Solution: x2 + y

2 = 80. (x – y)

2 =36 = 80 – 2xy.

xy = 44/2 = 22.

C. I have rs.500 in my pocket, I spent 2/5th of

money on a book, find the price of that book and

find the money left with me?

Solution: Cost of the book is 2/5 500 = 200.

Remaining money is 500 – 200 = 300 or 1 – 2/5

= 3/5, = 3/5500 = 300.

D. A person spends 1/3rd

of his salary on food, 1/4th

on clothes and 1/8th on house rent; now he left

with rs.1400, find his salary?

Solution: Remaining is 1 – (1/3) – (1/4) – (1/8) =

(24 – 8 – 6 –3)/24 = 7/24. (7/24)Total salary =

1400. Salary = 20 24 = rs.4800

E. If 21 is added to a number it becomes 7 less than

thrice of a number, and then the number is?

(S.S.C 2012) A 14 B 16 C 18 D 19

Solution: Let number is n. 21 + n = 3n7, n =

14.

F. Two pens and three pencils cost rs.86. Four pens

and a pencil cost rs.112. Find the cost of a pen

and that of a pencil?

Solution: Let the cost of pen is x rupees and

pencil cost is y rupees. 2x + 3y = 86. 4x + y =

112. By solving above equations, we get x =

rs.25 and y = rs.12.

27. 4 5

of of15 7

a number is greater than 4 2

of of9 5

of

the same number by 8. What is the half of that

number? A 630 B 540 C 315 D 100

28. A person spends 1/3rd

of his salary on food,

1/4thof remaining on clothes and 1/8

thof

remaining on house rent; now he left with

rs.1400, find his salary? A 3200 B 2400 C 4800

D 4000

29. In a basket, there are 125 flowers. A man goes to

worship and offers as many flowers at each

temple as there are temples in the city. Thus he

needs 5 baskets of flowers. Find the number of

temples in the city (S.S.C CGL 2012) A 27 B 25

C 26 D 24

30. A man divides rs.8600 among 5 sons, 4

daughters and 2 nephews. If each daughter

receives four times as much as each nephew, and

each son receives five times as much as each


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Number Systems 11


nephew, how much does each daughter receive?

(S.S.C CGL 2000) A 200 B 600 C 800 D 500

31. When an amount was distributed among 14

boys, each of them got rs.80 more than the

amount received by each boy when the same

amount is distributed equally among 18 boys.

What was the amount? A 4080 B 5040 C 1200

D 1040

32. Kiran had 85 currency notes in all, some of

which were of rs.100 denomination and

remaining of rs.50 denomination. The total

amount of all these currency notes was rs.5000.

How much amount did she have in the

denomination of rs.50? A 15 B 60 C 70 D None

33. The sum of a number and its reciprocal is

164/18. Find the sum of that number and its

square root? A 12 B 16 C 8 D 20

34. The product of three consecutive even numbers

when divided by 24 is 96; find the product of

their square roots? A 32 B 36 C 28 D 48

35. A total of 68 chocolates were distributed among

12 children such that each girl gets 5 chocolates

and each boy gets 6 chocolates, find the number

of boys? A 8 B 6 C 9 D 4

36. My friends were organizing a picnic. They

estimated the expenditure to be Rs.500, they

mobilized some more friends. The number of

people who actually went to the picnic increased

by five, the expenditure per head came down by

five rupees. The people went for the picnics

were?

(A) 35 (B) 15 (C) 20 (D) 25

37. If a number multiplied by 33 and the same

number is added to it, then we get a number that

is half the square of that number. Find the

number:

(A) 66 (B) 67 (C) 68 (D) None

38. The fuel in tank contains 1/5 of the total capacity

of the tank. When 22 more liters of fuel are

poured into the tank, the indicator rests at the

three-fourth of the full mark. Find the capacity

of the fuel tank.

(A) 36 L (B) 40 L (C) 44 L (D) 32 L

39. Out of a group of swans, 7/2 times the square

root of total number of swans are playing on the

shore of the pond. The remaining two are inside

the pond. Find the total number of swans?

(A) 12 (B) 36 (C) 16 (D) 15

40. The highest score in an innings is 2/9th of the

total score in the inning and the next highest

score is 2/9 of the remainder score. The

difference of two scores is 8 runs. Find the total

score in the innings?

(A) 160 (B) 162 (C) 186 (D) 196

41. Find the number of even numbers between 102

and 252 if:

A. Both the ends included

B. Only one end is included

C. Neither end is included

42. Find the number of numbers between 301 and

400 (both included), that are not divisible by 3.

(A) 68 (B) 65 (C) 66 (D) 67

43. The sum of the third powers of the first 100

natural numbers will have a unit’s digit of?

(A) 5 (B) 6 (C) 4 (D) 0

44. In a three digit number the sum of the digits is

13. The tens digit is one less than the units digit.

Which of the following numbers cannot be a

digit in the hundreds place?

(A) 4 (B) 2 (C) 5 (D) 6

45. When we multiply a certain two digit number by

the sum of its digits, 238 is achieved. If we

multiply the number written in reverse order of

the same digits by the sum of the digits, we get

301, find the number:

(A) 27 (B) 43 (C) 34 (D) 26

46. The unit’s digit of the expression 2139 137195 477 3333684 is:

(A) 2 (B) 4 (C) 5 (D) 0

47. Which of the following can never be in the

ending of a perfect square?


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Number Systems 12


(A) 5 (B) 9 (C) 00 (D) 000

48. The difference of 2510 7 and 2410 x is divisible

by 3, then x:

(A) 2 (B) 3 (C) 7 (D) 0

49. Find the two digit number if it is known that the

ratio of the required number and the sum of its

digits is 8 as also quotient of the product of its

digits and that of sum is 14/9:

(A) 42 (B) 72 (C) 86 (D) 91

50. Let x, y and z be distinct integers. x and y are

odd and positive, and z is even and positive.

Which one of the following statements cannot be

true?

(A) 2.x y z is even (B)

2.x z y is odd

(C) 2.x y z is even (D) 2.x y z is odd

Practice set -2 Key:

1- C 2 – B 3 – A 4 – B 5 – D 6 – B 7 – A 8 – C 9 –

Ans: 144; 10 – C 11 – D 12 – C 13 – C 14- B 15-

Ans: 51, 9; 16 – C 17- C 18 – C 19 - D 20 – C 21 –

22 – D 23 – 24 – D 25 – C 26 – D 27 – C 28 – A 29

– B 30 – C 31- B 32 – D 33 – A 34 – D 35- 36 – D

37 – A 38 – B 39 – C 40 –B 41 - Ans: 76, 75, 74

42 – D 43 – D 44 – C 45 – C 46 – D 47 – D 48 – A

49 – B 50 –D


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