The least integer when multiplied by 2940 becomes a perfect square is
10
15
20
35
Answer is Wrong!
Answer is Right!
This question was previously asked in
UPSC CAPF – 2019
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.