By which smallest positive integer should 5000 be divided so that the resulting quotient is a perfect square number?

Introduction / Context:
In this question, we are working with perfect squares and factorisation. The task is to find the smallest positive integer by which 5000 should be divided so that the quotient is a perfect square number. This tests understanding of prime factorisation and the condition for a number to be a perfect square in terms of its prime powers.

Given Data / Assumptions:

- Original number = 5000.

- We divide 5000 by a positive integer N.

- The quotient 5000 / N must be a perfect square.

- We want the least possible value of N.

Concept / Approach:
A positive integer is a perfect square if, in its prime factorisation, all prime exponents are even. So, we factorise 5000 into primes, then decide what should be divided out so that the remaining exponents are all even. The smaller the divisor N, the better, provided the quotient still becomes a perfect square.

Step-by-Step Solution:
Step 1: Factorise 5000. 5000 = 5 * 1000 = 5 * (10^3) = 5 * (2^3 * 5^3) = 2^3 * 5^4. Step 2: Check exponents of prime factors. We have 2^3 and 5^4. The exponent of 5 is 4 (even), but the exponent of 2 is 3 (odd). Step 3: For 5000 / N to be a perfect square, the exponents in the quotient must all be even. Let 5000 / N = 2^(3 - e2) * 5^(4 - e5), where N = 2^e2 * 5^e5 and e2, e5 are non negative integers. Step 4: We want 3 - e2 to be even and non negative. The smallest such choice is e2 = 1, giving 3 - 1 = 2, which is even. Step 5: Since 4 is already even, we can take e5 = 0, so the exponent of 5 remains 4. Step 6: Thus, the smallest divisor that works is N = 2^1 * 5^0 = 2. Step 7: Check: 5000 / 2 = 2500 = 50^2, which is a perfect square.

Verification / Alternative check:
Try the other options quickly: 5000 / 5 = 1000 (not a perfect square). 5000 / 10 = 500 (not a perfect square). 5000 / 15 ≈ 333.33 (not an integer, so cannot be a perfect square). This confirms that 2 is the only correct and smallest divisor among the given options.

Why Other Options Are Wrong:
- 5: Leaves 1000, which is not a perfect square.
- 10: Leaves 500, which is not a perfect square.
- 15: Does not even give an integer quotient, so it is invalid for forming a perfect square.

Common Pitfalls:
Students sometimes mistakenly try to make 5000 itself a perfect square by multiplying it instead of considering division as asked. Others may guess values without fully factorising 5000 and checking exponents of primes, which can easily lead to wrong answers. Another error is ignoring that the quotient must be an integer perfect square, not just a square of a fraction.

Final Answer:
Thus, the smallest integer by which 5000 should be divided to obtain a perfect square is 2.