The least integer when multiplied by 2940 becomes a perfect square is

Answer is Right!

Answer is Wrong!

This question was previously asked in

UPSC CAPF – 2019

The correct answer is B) 15.

To find the least integer by which 2940 must be multiplied to make it a perfect square, we need to find the prime factorization of 2940.
2940 = 294 * 10
294 = 2 * 147 = 2 * 3 * 49 = 2 * 3 * 7^2
10 = 2 * 5
So, 2940 = (2 * 3 * 7^2) * (2 * 5) = 2^2 * 3^1 * 5^1 * 7^2.
For a number to be a perfect square, the exponents of all prime factors in its prime factorization must be even.
In the prime factorization of 2940 (2^2 * 3^1 * 5^1 * 7^2), the exponents of the prime factors are 2 (for 2), 1 (for 3), 1 (for 5), and 2 (for 7).
The exponents of 3 and 5 are odd. To make them even, we need to multiply 2940 by 3^1 and 5^1.
The least integer needed to multiply is 3 * 5 = 15.
When multiplied by 15, the number becomes 2^2 * 3^1 * 5^1 * 7^2 * (3 * 5) = 2^2 * 3^2 * 5^2 * 7^2 = (2 * 3 * 5 * 7)^2. This is a perfect square.

A number is a perfect square if and only if all the exponents in its prime factorization are even. To find the least multiplier to make a number a perfect square, identify the prime factors with odd exponents and multiply the number by the product of these prime factors raised to the power needed to make their exponents even (which will always be 1 for each such prime factor).

More similar MCQ questions for Exam preparation:-