4GMAT Diagnostic Test Q14 - Problem Solving - Coordinate Geometry
4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF
Transcript of 4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF
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GMAT QUANTITATIVE REASONING
NUMBER PROPERTIES : HCF
PROBLEM SOLVING
Diagnostic Test
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Question
A bag contains 72 red marbles, 45 green marbles and 108 blue marbles. These are packed into packets containing equal number of marbles of the same colour. What is minimum number of packets required?A. 9B. 36C. 25D. 19E. 21
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Step 1
Decoding the information given
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What is minimum number of packets required?
72 red marbles, 45 green marbles and 108 blue marbles.
Idea #1
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What is minimum number of packets required?
72 red marbles, 45 green marbles and 108 blue marbles.
Idea #1Packed into packets
containing equal number
of marbles of the
same color.
![Page 6: 4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF](https://reader033.fdocuments.us/reader033/viewer/2022042608/55b7f83fbb61eb086d8b45f7/html5/thumbnails/6.jpg)
What is minimum number of packets required?
72 red marbles, 45 green marbles and 108 blue marbles.
Idea #1Packed into packets
containing equal number
of marbles of the
same color.
Equal Number in Each Packet
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What is minimum number of packets required?
72 red marbles, 45 green marbles and 108 blue marbles.
Idea #1Packed into packets
containing equal number
of marbles of the
same color.
Equal Number in Each PacketNumber of marbles in all the packets is the same. Let us say the number is ‘n’
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What is minimum number of packets required?
72 red marbles, 45 green marbles and 108 blue marbles.
Idea #1Packed into packets
containing equal number
of marbles of the
same color.
Equal Number in Each Packet
Same color
Number of marbles in all the packets is the same. Let us say the number is ‘n’
![Page 9: 4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF](https://reader033.fdocuments.us/reader033/viewer/2022042608/55b7f83fbb61eb086d8b45f7/html5/thumbnails/9.jpg)
What is minimum number of packets required?
72 red marbles, 45 green marbles and 108 blue marbles.
Idea #1Packed into packets
containing equal number
of marbles of the
same color.
Equal Number in Each Packet
All ‘n’ marbles in a packet should be of the same colour
Same color
Number of marbles in all the packets is the same. Let us say the number is ‘n’
![Page 10: 4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF](https://reader033.fdocuments.us/reader033/viewer/2022042608/55b7f83fbb61eb086d8b45f7/html5/thumbnails/10.jpg)
What is minimum number of packets required?
72 red marbles, 45 green marbles and 108 blue marbles.
Idea #1Packed into packets
containing equal number
of marbles of the
same color.
Equal Number in Each Packet
All ‘n’ marbles in a packet should be of the same colour
Same color
What does it translate into?
Number of marbles in all the packets is the same. Let us say the number is ‘n’
![Page 11: 4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF](https://reader033.fdocuments.us/reader033/viewer/2022042608/55b7f83fbb61eb086d8b45f7/html5/thumbnails/11.jpg)
What is minimum number of packets required?
72 red marbles, 45 green marbles and 108 blue marbles.
Idea #1Packed into packets
containing equal number
of marbles of the
same color.
Equal Number in Each Packet
All ‘n’ marbles in a packet should be of the same colour
Same color
What does it translate into?
‘n’ should be a factor of 72 if all red marbles are to be packed into packets containing ‘n’ marbles.
Number of marbles in all the packets is the same. Let us say the number is ‘n’
![Page 12: 4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF](https://reader033.fdocuments.us/reader033/viewer/2022042608/55b7f83fbb61eb086d8b45f7/html5/thumbnails/12.jpg)
What is minimum number of packets required?
72 red marbles, 45 green marbles and 108 blue marbles.
Idea #1Packed into packets
containing equal number
of marbles of the
same color.
Equal Number in Each Packet
All ‘n’ marbles in a packet should be of the same colour
Same color
What does it translate into?
‘n’ should be a factor of 72 if all red marbles are to be packed into packets containing ‘n’ marbles.
By the same token, n should be a factor of 45 and 108 as well.
Number of marbles in all the packets is the same. Let us say the number is ‘n’
![Page 13: 4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF](https://reader033.fdocuments.us/reader033/viewer/2022042608/55b7f83fbb61eb086d8b45f7/html5/thumbnails/13.jpg)
What is minimum number of packets required?
72 red marbles, 45 green marbles and 108 blue marbles.
Idea #1Packed into packets
containing equal number
of marbles of the
same color.
Equal Number in Each Packet
All ‘n’ marbles in a packet should be of the same colour
Same color
What does it translate into?
‘n’ should be a factor of 72 if all red marbles are to be packed into packets containing ‘n’ marbles.
By the same token, n should be a factor of 45 and 108 as well.
‘n’ is a common factor of 72, 45 and 108.
Number of marbles in all the packets is the same. Let us say the number is ‘n’
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Idea #2
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Idea #2What is the minimum
number of packets
required?
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Idea #2What is the minimum
number of packets
required?
Number of packets - MINIMUM
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Idea #2What is the minimum
number of packets
required?
Number of packets - MINIMUMMore the number of marbles in each packet, lesser the number of packets. i.e., maximize ‘n’ to minimize number of packets.
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Idea #2What is the minimum
number of packets
required?
Number of packets - MINIMUMMore the number of marbles in each packet, lesser the number of packets. i.e., maximize ‘n’ to minimize number of packets.
‘n’ is a common factor
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Idea #2What is the minimum
number of packets
required?
Number of packets - MINIMUMMore the number of marbles in each packet, lesser the number of packets. i.e., maximize ‘n’ to minimize number of packets.
We determined that ‘n’ is a factor common to 72, 45, and 108
‘n’ is a common factor
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Idea #2What is the minimum
number of packets
required?
Number of packets - MINIMUMMore the number of marbles in each packet, lesser the number of packets. i.e., maximize ‘n’ to minimize number of packets.
We determined that ‘n’ is a factor common to 72, 45, and 108
‘n’ is a common factor
Combining findings above
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Idea #2What is the minimum
number of packets
required?
Number of packets - MINIMUMMore the number of marbles in each packet, lesser the number of packets. i.e., maximize ‘n’ to minimize number of packets.
We determined that ‘n’ is a factor common to 72, 45, and 108
‘n’ is a common factor
Combining findings above
‘n’ should be a common factor of 72, 45, and 108 and ‘n’ has to be as high as possible.
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Idea #2What is the minimum
number of packets
required?
Number of packets - MINIMUMMore the number of marbles in each packet, lesser the number of packets. i.e., maximize ‘n’ to minimize number of packets.
We determined that ‘n’ is a factor common to 72, 45, and 108
‘n’ is a common factor
Combining findings above
‘n’ should be a common factor of 72, 45, and 108 and ‘n’ has to be as high as possible.
‘n’ is the highest common factor (HCF) of 72, 45 and 108.
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Step 2
How to compute HCF?
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Prime Factorization Method
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Prime Factorization Method
STEP
01Prime factorize the 3 numbers
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Prime Factorization Method
STEP
01Prime factorize the 3 numbers
72 = 23 * 32
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Prime Factorization Method
STEP
01Prime factorize the 3 numbers
72 = 23 * 32
45 = 32 * 5
![Page 28: 4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF](https://reader033.fdocuments.us/reader033/viewer/2022042608/55b7f83fbb61eb086d8b45f7/html5/thumbnails/28.jpg)
72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Prime Factorization Method
STEP
01Prime factorize the 3 numbers
72 = 23 * 32
45 = 32 * 5
108 = 22 * 33
![Page 29: 4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF](https://reader033.fdocuments.us/reader033/viewer/2022042608/55b7f83fbb61eb086d8b45f7/html5/thumbnails/29.jpg)
72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Prime Factorization Method
STEP
01Prime factorize the 3 numbers
72 = 23 * 32
45 = 32 * 5
108 = 22 * 33
STEP
02List down prime factors
common to all the numbers
![Page 30: 4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF](https://reader033.fdocuments.us/reader033/viewer/2022042608/55b7f83fbb61eb086d8b45f7/html5/thumbnails/30.jpg)
72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Prime Factorization Method
STEP
01Prime factorize the 3 numbers
72 = 23 * 32
45 = 32 * 5
108 = 22 * 33
STEP
02List down prime factors
common to all the numbers
![Page 31: 4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF](https://reader033.fdocuments.us/reader033/viewer/2022042608/55b7f83fbb61eb086d8b45f7/html5/thumbnails/31.jpg)
72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Prime Factorization Method
STEP
01Prime factorize the 3 numbers
72 = 23 * 32
45 = 32 * 5
108 = 22 * 33
STEP
02List down prime factors
common to all the numbers
3 is the only prime factor common
to all the numbers
![Page 32: 4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF](https://reader033.fdocuments.us/reader033/viewer/2022042608/55b7f83fbb61eb086d8b45f7/html5/thumbnails/32.jpg)
72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Prime Factorization Method
STEP
01Prime factorize the 3 numbers
72 = 23 * 32
45 = 32 * 5
108 = 22 * 33
STEP
02List down prime factors
common to all the numbers
3 is the only prime factor common
to all the numbers
Pick the lowest power of the
prime factors common to all
numbers and multiply to get
the HCF
STEP
03
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Prime Factorization Method
STEP
01Prime factorize the 3 numbers
72 = 23 * 32
45 = 32 * 5
108 = 22 * 33
STEP
02List down prime factors
common to all the numbers
3 is the only prime factor common
to all the numbers
Pick the lowest power of the
prime factors common to all
numbers and multiply to get
the HCF
The lowest power of the only
common factor ‘3’ is 32STEP
03
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Prime Factorization Method
STEP
01Prime factorize the 3 numbers
72 = 23 * 32
45 = 32 * 5
108 = 22 * 33
STEP
02List down prime factors
common to all the numbers
3 is the only prime factor common
to all the numbers
Pick the lowest power of the
prime factors common to all
numbers and multiply to get
the HCF
The lowest power of the only
common factor ‘3’ is 32
The HCF is 32 = 9
STEP
03
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It’s far from over
We have to compute the number of packets
– not the number of marbles in each packet
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Number of marbles in each packet = 9
![Page 37: 4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF](https://reader033.fdocuments.us/reader033/viewer/2022042608/55b7f83fbb61eb086d8b45f7/html5/thumbnails/37.jpg)
72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Number of marbles in each packet = 9
72 red marbles can therefore, be packed into 8 packets.
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Number of marbles in each packet = 9
72 red marbles can therefore, be packed into 8 packets.
45 green marbles can be packed into 5 packets.
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72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Number of marbles in each packet = 9
72 red marbles can therefore, be packed into 8 packets.
45 green marbles can be packed into 5 packets.
108 blue marbles can be packed into 12 packets.
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72 red marbles, 45 green marbles and 108 blue marbles.
Total number of packets = 8 + 5 + 12 = 25 packets
What is minimum number of packets required?
Number of marbles in each packet = 9
72 red marbles can therefore, be packed into 8 packets.
45 green marbles can be packed into 5 packets.
108 blue marbles can be packed into 12 packets.
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72 red marbles, 45 green marbles and 108 blue marbles.
Total number of packets = 8 + 5 + 12 = 25 packets
What is minimum number of packets required?
Number of marbles in each packet = 9
72 red marbles can therefore, be packed into 8 packets.
45 green marbles can be packed into 5 packets.
108 blue marbles can be packed into 12 packets.
Choice C is the correct answer
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