# Geometry Puzzle – Critical Thinking 19

*Critical thinking* involves mindful communication. To develop your creative thinking, use this kind of math puzzles. The solution is also provided.

## Challenge:

N and P are prime numbers. How many divisors N\(^2\) **×** P\(^4\) has?

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The correct answer is 15.

If p and q are prime numbers, then for p\(^m\)q\(^n\), the number of divisors is (m+1)(n+1).

Thus the number of divisors for N\(^2\) × P\(^4\) is (2+1)(4+1) = 15

Another way to solve this problem is by providing numbers for N and P. Let’s put 2 for N and 3 for P. Then, the value of N\(^2\) × P\(^4\) is 2\(^2\) × 3\(^4\) = 324

Now, find the factors of 324.

324: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324

324 have 15 factors.

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