By what least positive integer should 1200 be multiplied so that the product becomes a perfect square?

Introduction / Context:
This problem focuses on perfect squares and prime factorisation. We are asked to find the smallest integer by which 1200 must be multiplied to obtain a perfect square. This requires understanding how prime exponents must behave in the factorisation of a perfect square.

Given Data / Assumptions:

- Original number = 1200.

- We multiply 1200 by a positive integer k.

- The product 1200 * k must be a perfect square.

- We want the smallest such k.

Concept / Approach:
A positive integer is a perfect square if, in its prime factorisation, all the exponents are even. So we factorise 1200 into primes, look at the parity (even or odd) of each exponent, and then decide what smallest factor k must be added so that all exponents in the product 1200 * k are even.

Step-by-Step Solution:
Step 1: Factorise 1200. 1200 = 12 * 100. 12 = 2^2 * 3, and 100 = 2^2 * 5^2. So 1200 = (2^2 * 3) * (2^2 * 5^2) = 2^4 * 3^1 * 5^2. Step 2: Examine the exponents of primes. Exponent of 2 is 4 (even). Exponent of 3 is 1 (odd). Exponent of 5 is 2 (even). Step 3: For a perfect square, all exponents must be even. Only the exponent of 3 is odd, equal to 1. Step 4: Multiply 1200 by the smallest factor that makes exponent of 3 even. Multiplying by 3 changes the exponent of 3 from 1 to 2. So we take k = 3. Step 5: Check the new factorisation. 1200 * 3 = 3600. 3600 = 36 * 100 = 6^2 * 10^2 = (6 * 10)^2 = 60^2, which is a perfect square.

Verification / Alternative check:
We can quickly check other options: 1200 * 2 = 2400, not a perfect square. 1200 * 4 = 4800, not a perfect square. 1200 * 5 = 6000, not a perfect square. Therefore, only k = 3 leads to a perfect square, and it is the smallest such value.

Why Other Options Are Wrong:
- 2: Leaves the exponent of 3 unchanged at 1, so the product is not a perfect square.
- 4: Changes only powers of 2, not 3, so exponent of 3 is still 1.
- 5: Changes exponent of 5 but exponent of 3 remains odd.

Common Pitfalls:
A frequent mistake is to try to guess the multiplier instead of factorising the number. Some learners also think in terms of approximate square roots without checking the prime factorisation, which can be misleading. Working directly with prime exponents is the most reliable approach.

Final Answer:
The least integer by which 1200 should be multiplied is 3.