What will be the remainder when 2384 is divided by 17?

Concept: When a large power an is divided by a modulus m, look for a residue of the base that simplifies under that power — in particular, a base congruent to -1 (mod m), since (-1) raised to an even exponent equals 1 and to an odd exponent equals m-1. This turns an unwieldy exponentiation into a simple parity check on the exponent.

1. The modulus is 17. Compute 24 = 16, and note that 16 ≡ -1 (mod 17), since 17 - 16 = 1.

2. Write the exponent 384 as a multiple of 4: 384 = 4 × 96, so 2384 = (24)96 = 1696.

3. Substitute the congruence 16 ≡ -1 (mod 17): 1696 ≡ (-1)96 (mod 17).

4. Since 96 is an even number, (-1)96 = 1.

5. Therefore 2384 ≡ 1 (mod 17), so the remainder when 2384 is divided by 17 is 1.

Cross-check: Independently, by Fermat's Little Theorem, since 17 is prime and 2 is not divisible by 17, 216 ≡ 1 (mod 17). As 384 = 16 × 24 is a multiple of 16, 2384 = (216)24 ≡ 124 = 1 (mod 17) — the same result obtained through a different reduction, confirming the remainder is 1.