Question: The sum of two positive integers is 156 and their highest common factor is 13. How many pairs of such numbers are possible?

1. 1
2. 2
3. 3
4. 4
5. 5

Correct Answer: B
Solution and Explanation:
Approach Solution 1:

First number= 13a
The second number is 13b, and is co-prime (roughly prime), hence their HCF is 1.
13a+13b = 156
i.e. a+b = 12

i.e (a,b) = (1, 11) (2, 10) (3, 9) (4, 8) (5, 7) (6, 6)
i.e. only two such pairs
Hence the correct answer is B.

Approach Solution 2:

There is just one component that unites two numbers, and that factor is 13, hence two numbers only have 13 in common.
Thus, let 13A and 13B be the two numbers.
13(A+B)=156

A+B=12
As previously stated, A and B will now be co-primes (have no common factor other than 1)
A and B might therefore have values of (1,11) and (5,7).
The maximum number of pairings is two.

Hence the correct answer is B.

Approach Solution 3:

As , 13 is the highest common factor of two number and sum of 2 is 156,
Therefore, 13x+13y = 156
13(x+y) = 156 ------ 1
x+y = 12 ( where x & y has to be co-primes)

Now we have to check values for x & y.
If x = 1, then y = 11 ; put in 1, so the sum is 156.
If x= 5 , then y = 7 ; put in 1, so the sum is 156.

So only 2 pairs are possible.
Hence the correct answer is B.

“The sum of two positive integers is 156 and their highest common factor is 13. How many pairs of such numbers are possible?” - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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