4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF

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



Idea #1Packed into packets

containing equal number

of marbles of the

same color.





of marbles of the

same color.

Equal Number in Each Packet





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’





of marbles of the

same color.


Same color

Number of marbles in all the packets is the same. Let us say the number is ‘n’





of marbles of the

same color.


All ‘n’ marbles in a packet should be of the same colour

Same color






of marbles of the

same color.



Same color

What does it translate into?






of marbles of the

same color.



Same color


‘n’ should be a factor of 72 if all red marbles are to be packed into packets containing ‘n’ marbles.






of marbles of the

same color.



Same color



By the same token, n should be a factor of 45 and 108 as well.






of marbles of the

same color.



Same color



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.




Idea #2



Idea #2What is the minimum

number of packets

required?




number of packets

required?

Number of packets - 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.




number of packets

required?


‘n’ is a common factor




number of packets

required?


We determined that ‘n’ is a factor common to 72, 45, and 108





number of packets

required?




Combining findings above




number of packets

required?





‘n’ should be a common factor of 72, 45, and 108 and ‘n’ has to be as high as possible.




number of packets

required?





‘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?



Prime Factorization Method




STEP

01Prime factorize the 3 numbers




STEP


72 = 23 * 32




STEP


72 = 23 * 32

45 = 32 * 5




STEP


72 = 23 * 32

45 = 32 * 5

108 = 22 * 33




STEP


72 = 23 * 32

45 = 32 * 5

108 = 22 * 33

STEP

02List down prime factors

common to all the numbers




STEP


72 = 23 * 32

45 = 32 * 5

108 = 22 * 33

STEP



3 is the only prime factor common

to all the numbers




STEP


72 = 23 * 32

45 = 32 * 5

108 = 22 * 33

STEP




to all the numbers

Pick the lowest power of the

prime factors common to all

numbers and multiply to get

the HCF

STEP

03




STEP


72 = 23 * 32

45 = 32 * 5

108 = 22 * 33

STEP




to all the numbers




the HCF

The lowest power of the only

common factor ‘3’ is 32STEP

03




STEP


72 = 23 * 32

45 = 32 * 5

108 = 22 * 33

STEP




to all the numbers




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



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.


Total number of packets = 8 + 5 + 12 = 25 packets







Total number of packets = 8 + 5 + 12 = 25 packets






Choice C is the correct answer

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4GMAT Diagnostic Test Q1 - Problem Solving : Number Properties HCF

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