Which one of the following is the smallest number by which 2880 must b

Which one of the following is the smallest number by which 2880 must be divided in order to make it a perfect square ?

Answer is Right!

Answer is Wrong!

This question was previously asked in

UPSC CAPF – 2017

To find the smallest number by which 2880 must be divided to make it a perfect square, we first find the prime factorization of 2880.
2880 = 288 * 10 = (144 * 2) * (2 * 5) = (12² * 2) * (2 * 5) = ((2² * 3)²) * 2² * 5 = (2⁴ * 3²) * 2² * 5 = 2⁶ * 3² * 5¹
For a number to be a perfect square, the exponents of all its prime factors must be even. In the factorization 2⁶ * 3² * 5¹, the exponents are 6 (even), 2 (even), and 1 (odd).
To make the exponent of 5 even (which is 1), we must divide by 5 raised to the power of its odd exponent, i.e., 5¹.
So, we must divide 2880 by 5.
2880 / 5 = 576.
Let’s check if 576 is a perfect square: 576 = 24 * 24 = 24². Yes, it is.
Therefore, the smallest number by which 2880 must be divided is 5.

A number is a perfect square if and only if the exponents of all prime factors in its prime factorization are even. To make a number a perfect square by division, divide by the product of prime factors with odd exponents, each raised to the power of their odd exponent.

If the question asked for the smallest number to *multiply* by, you would multiply by the product of the prime factors with odd exponents, each raised to the power needed to make the exponent even (which is the odd exponent itself). In this case, multiply by 5¹.

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