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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