The Greatest Common Divisor of Two Positive Integers is 25 GMAT Problem Solving

Question: The greatest common divisor of two positive integers is 25. If the sum of the integers is 350, then how many such pairs are possible?

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

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

The problem statement states that:

Given:

• The greatest common divisor of two positive integers is 25.
• The sum of the integers is 350.

Find Out:

• The number of such possible pairs.

We are told that the greatest common factor of two integers is 25.
Therefore, these integers are 25x and 25y, for some positive integers x and y.
It is required to notice that x and y must not share any common factor but 1. This is because if they do, then the GCF of 25x and 25y will be more than 25.

Next, we know that 25x + 25y = 350.
Reducing by 25 gives x + y = 14.

Now, since x and y do not share any common factor but 1 then (x, y) can be only (1, 13), (3, 11) or (5, 9).
All other pairs (2, 12), (4, 10), (6, 8) and (7, 7) do share a common factor greater than 1.

Therefore, there are only three pairs of such numbers possible:
25 ∗ 1 = 25 and 25 ∗ 13 = 325
25 ∗ 3 = 75 and 25 ∗ 11 = 275;
25 ∗ 5 = 125 and 25 ∗ 9 = 225.
Hence, the number of such possible pairs = 3.

Approach Solution 2:

The problem statement suggests that:

Given:

• The greatest common divisor of two positive integers is 25.
• The sum of the integers is 350.

Find Out:

• The number of such possible pairs.

Let the two numbers be x1 and x2,
Since their GCF is 25, they must be multiples of 25.
Therefore, 25 * x1, 25 * x2, given their sum is 350

Hence, by solving, we get:
x1+x2 = 14
Now there are many possibilities for this equation, but we have to see for numbers which don't share a common factor. Otherwise, we will end up changing the greatest common divisor given in the question.

Therefore, possible values are (1,13) (3,11) (5,9)
Hence, the number of such possible pairs = 3.

Approach Solution 3:

The problem statement indicates that:

Given:

• The greatest common divisor of two positive integers is 25.
• The sum of the integers is 350.

Find Out:

• The number of such possible pairs.

HCF = 25
Sum of two integers = 350

Let's solve the problem by using the concept:
If the number is 25a and 25b, then a and b will be co-prime numbers.

Let the number be 25a and 25b.
Then as per the conditions of the question, we know that:
25a + 25b = 350
=> 25(a+b) = 350
=> a + b = 14

Now, let’s find out the co-prime pair whose sum is 14.
(1,13) (3,11) (5,9)
Hence, the number of such possible pairs = 3.

“The greatest common divisor of two positive integers is 25”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Guide 2021”. To solve the GMAT Problem Solving questions, the candidates must have basic knowledge of mathematics. The candidates can follow GMAT Quant practice papers to practise varieties of questions that will enable them to improve their mathematical understanding.

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