Problem 24 · 2010 Math Kangaroo
Stretch
Number Theory
perfect-squarecasework
For how many integers n with \(1 \le n \le 100\) is \(n^n\) a square number?
Show answer
Answer: C — 55
Show hints
Hint 1 of 2
If n is even, the exponent n is even, so n to the n is automatically a square.
Still stuck? Show hint 2 →
Hint 2 of 2
For odd n, n to the n is a square only when n itself is a perfect square.
Show solution
Approach: split by parity of n
- For even n the power has an even exponent, so it is always a perfect square: 50 values.
- For odd n, n to the n is a square only if n is a square: 1, 9, 25, 49, 81 - 5 values.
- Total = 50 + 5 = 55.
Mark:
· log in to save