GMAT QUANTITATIVE REASONING
NUMBER PROPERTIES : HCF
PROBLEM SOLVING
Diagnostic Test
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
Step 1
Decoding the information given
What is minimum number of packets required?
72 red marbles, 45 green marbles and 108 blue marbles.
Idea #1
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.
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
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’
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’
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’
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’
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’
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’
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’
72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Idea #2
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?
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
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.
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
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
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
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.
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.
Step 2
How to compute HCF?
72 red marbles, 45 green marbles and 108 blue marbles.
What is minimum number of packets required?
Prime Factorization Method
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 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
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
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
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
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
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
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
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
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
It’s far from over
We have to compute the number of packets
– not the number of marbles in each packet
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, 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.
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.
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.
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.
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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