If y is the Smallest Positive Integer Such That 3,150 Multiplied By y is the Square of an Integer GMAT Problem Solving

Question: If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be

1. 2
2. 5
3. 6
4. 7
5. 14

“If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide 2022”. 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.

Solution and Explanation

Approach Solution 1:

This problem can be solved by expressing a perfect square and its prime factors.
The prime factor of 3150 = 2x3x3x5x5x7
The square root of 3150 =\(\sqrt{2*3*3*5*5*7}=3*5\sqrt{2*7}\)
To make 3150 a perfect square minimum number, we need to multiply 2x7 =14

Correct Answer: E

Approach Solution 2:

This specific problem can be solved by expressing a perfect square(greater than 1) by its unique prime factors. A prime factor is supposedly an even number.
Factoring 3,150 we get,
3,150 = 315 x 10 = 5 x 63 x 10 = 5 x 7 x 3 x 3 x 5 x 2
3,150 = 2^1 x 3^2 x 5^2 x 7^1

(Note: The exponents of both 2 and 7 are odd numbers. This states that 3,150 itself is not a perfect square.)
It is given that 3,150 multiplied by y is the square of an integer. We can write this as

\(2^1*3^2*5^2*7^1*y\)= square of an integer
According to the rule of unique prime factors' exponents. All the components must be even numbers. Therefore, we get one more 2 and one more 7. Therefore, y = 7 x 2 = 14.

Correct Answer: E

Approach Solution 3:

Key concept: The prime factorization of a perfect square (the square of an integer) will have an EVEN number of each prime.
For example, 36 = (2)(2)(3)(3)
And 400 = (2)(2)(2)(2)(5)(5)

Likewise, 3150y must have an EVEN number of each prime in its prime factorization.
So, 3150y = (2)(3)(3)(5)(5)(7)y
We have an EVEN number of 3's and 5's, but we have a single 2 and a single 7.
If y = (2)(7), then we get a perfect square.

That is: 3150y = (2)(2)(3)(3)(5)(5)(7)(7)

So, if y = 14, then 3150y is a perfect square.

Correct Answer: E

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