What is the smallest number by which 2880 must be divided so that the result is a perfect square?

Introduction / Context:
This question involves prime factorisation and the condition for a number to be a perfect square. A perfect square has all prime factors raised to even powers. You are asked to find the smallest positive integer that can divide 2880 so that the quotient becomes a perfect square. This is a common type of question in number theory sections of aptitude tests.

Given Data / Assumptions:

• The given number is 2880.
• We must divide 2880 by a positive integer k so that 2880 / k is a perfect square.
• We seek the smallest such k.

Concept / Approach:
First factor 2880 into prime factors and express it as a product of prime powers. For the quotient to be a perfect square, every prime in its factorisation must have an even exponent. Dividing by k will remove some prime factors from 2880. The strategy is to remove the minimal prime factor power that turns all exponents even. We then check which small option matches that required k.

Step-by-Step Solution:
Factor 2880: start with 2880 = 288 * 10. Factor 288 = 32 * 9 = 2^5 * 3^2. Factor 10 = 2 * 5. So 2880 = 2^5 * 3^2 * 2 * 5 = 2^6 * 3^2 * 5. Write 2880 as 2^6 * 3^2 * 5^1. For a perfect square, the exponents of all primes must be even. Here the exponents of 2 and 3 are 6 and 2, both even, but the exponent of 5 is 1, which is odd. To make all exponents even, we need to remove one factor of 5 by dividing by 5. Then the quotient is (2^6 * 3^2 * 5) / 5 = 2^6 * 3^2. This simplifies to (2^3 * 3)^2 = 24^2, which is a perfect square.

Verification / Alternative check:
Compute the quotient explicitly: 2880 / 5 = 576. Now check whether 576 is a perfect square. We know 24 * 24 = 576, so 576 is indeed a perfect square. If we tried a smaller divisor, for example 2 or 3, the quotient would still have an odd exponent on one of the prime factors and would not be a perfect square. Thus 5 is the smallest divisor that works.

Why Other Options Are Wrong:

• 2: 2880 / 2 = 1440, whose prime factorisation still includes 5^1, so it is not a perfect square.
• 3: 2880 / 3 = 960, which has 2^6 * 3^1 * 5, with an odd exponent on 3.
• 4: 2880 / 4 = 720, with factorisation 2^4 * 3^2 * 5, still having 5^1.
• 6: 2880 / 6 = 480, with prime factorisation 2^5 * 3^1 * 5, again not all exponents even.

Common Pitfalls:
Some students try to test each option by long division and then guess whether the quotient is a square, which is slow and error prone. Others forget that only the parity (evenness or oddness) of each exponent matters. A systematic approach is to factor the original number, look at exponents of each prime, and then decide which minimal prime powers must be removed or added. This technique applies to both making perfect squares and perfect cubes.

Final Answer:
The smallest number by which 2880 must be divided to obtain a perfect square is 5.