Remainder problem — Determine the remainder when 4^1000 is divided by 7. Identify and use the exponent cycle modulo 7.

Introduction / Context:
For remainders with large exponents, spotting cycles modulo a small base is the standard trick. The base 4 modulo 7 repeats with a short period due to Euler's theorem and orders of elements, allowing a quick reduction of the exponent 1000.

Given Data / Assumptions:

• Compute 4^1000 mod 7.
• Know φ(7) = 6 and that powers of 4 modulo 7 cycle.
• We need only the final remainder.

Concept / Approach:
Since 7 is prime, 4^6 ≡ 1 (mod 7). Therefore reduce 1000 modulo 6 to find the equivalent smaller exponent. Then compute 4^r mod 7 for the remainder exponent r and read off the result.

Step-by-Step Solution:
1) Because 4^6 ≡ 1 (mod 7), compute 1000 mod 6.2) 1000 = 6*166 + 4 → remainder exponent r = 4.3) Compute 4^4 mod 7: 4^2 = 16 ≡ 2 (mod 7), then 4^4 = (4^2)^2 ≡ 2^2 = 4 (mod 7).4) Hence, 4^1000 ≡ 4 (mod 7).

Verification / Alternative check:
Direct cycle: 4, 2, 1, 4, 2, 1, … every 3 steps within the 6-length residue pattern also shows the 4th term equals 4, confirming the result.

Why Other Options Are Wrong:
1 or 2 would require 1000 to land on different positions in the cycle; 0 is impossible since 7 ∤ 4; “None of these” is invalid because 4 fits perfectly.

Common Pitfalls:
Reducing 1000 modulo 7 instead of modulo 6; arithmetic slips computing 4^4; assuming the remainder must be small like 1 by default.

Final Answer:
4