30^72^87 divided by 11 gives remainder ?

Solution summary: compute 30^(72^87) modulo 11.

1. Step 1: Reduce the base modulo 11. 30 ≡ 8 (mod 11), so the expression equals 8^(72^87) (mod 11).

2. Step 2: Reduce the exponent modulo 10 (Euler's theorem). Since gcd(8,11)=1, exponents can be reduced mod ϕ(11)=10. Compute 72^87 (mod 10).

   Note 72 ≡ 2 (mod 10), so 72^87 ≡ 2^87 (mod 10). Powers of 2 modulo 10 cycle every 4, and 87 ≡ 3 (mod 4), so 2^87 ≡ 2^3 ≡ 8 (mod 10).

3. Step 3: Compute 8^8 modulo 11. Reduce 8 modulo 11 as -3: (-3)^8 = 3^8.

   Compute 3^8 using 3^5 ≡ 1 (mod 11): 3^8 = 3^5·3^3 ≡ 1·27 ≡ 27 ≡ 5 (mod 11).

Conclusion: the remainder when 30^(72^87) is divided by 11 is 5.