What is the remainder of (32^31^301) when it is divided by 9?

Step 1: Reduce the base modulo 9. 32 ≡ 5 (mod 9), so the expression becomes 5^{31^{301}} (mod 9).

Step 2: Use Euler's theorem. φ(9)=6, and gcd(5,9)=1, so 5^6 ≡ 1 (mod 9).

Step 3: Reduce the exponent modulo 6. 31 ≡ 1 (mod 6), so 31^{301} ≡ 1^{301} ≡ 1 (mod 6).

Step 4: Conclude the remainder. Therefore 5^{31^{301}} ≡ 5^{1} ≡ 5 (mod 9), so the remainder is 5.