IPMAT Indore 2025 · Quantitative Ability · Question 3
IPMAT Indore 2025 previous year question on Number of Factors, with the answer and a step-by-step solution.
The number of factors of {3}^{5}\times{5}^{8}\times{7}^{2} that are perfect squares is
Answer & solution
Answer: 30
Step 1 — what a factor looks like. Every factor has the form
{3}^{a}\times{5}^{b}\times{7}^{c}, 0\le a\le 5, 0\le b\le 8, 0\le c\le 2
Step 2 — what makes it a perfect square. A number is a perfect square exactly when every exponent in its prime factorisation is even. So we need a, b, c all even.
Step 3 — count the even choices for each exponent.
Prime | Exponent range | Even values allowed | Count |
|---|---|---|---|
3 | 0–5 | 0, 2, 4 | 3 |
5 | 0–8 | 0, 2, 4, 6, 8 | 5 |
7 | 0–2 | 0, 2 | 2 |
Step 4 — multiply (the three choices are independent):
3\times 5\times 2=30
Answer: 30
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