Remainder problem — Compute the remainder when 9^19 + 6 is divided by 8. Use modular reduction of the base.

Introduction / Context:We are asked for a remainder modulo 8. Reducing bases modulo 8 and recognizing simple patterns allow quick answers for large exponents. This is a standard exercise in modular arithmetic fluency for aptitude tests.

Given Data / Assumptions:

• Expression: 9^19 + 6.
• Modulus: 8.
• Basic rules: reduce the base first, then apply exponent laws.

Concept / Approach:Since 9 ≡ 1 (mod 8), any power 9^k ≡ 1^k ≡ 1 (mod 8). Then just add the remaining constant 6 and reduce modulo 8.

Step-by-Step Solution:1) Reduce base: 9 ≡ 1 (mod 8).2) Raise to power: 9^19 ≡ 1^19 = 1 (mod 8).3) Add 6: 1 + 6 = 7.4) Therefore (9^19 + 6) mod 8 = 7.

Verification / Alternative check:Because 9 ≡ 1 (mod 8) exactly, no cycle tracking is needed. Any exponent would still collapse to 1 before adding 6, confirming the same remainder.

Why Other Options Are Wrong:2, 3, 5, and 1 do not match the straightforward reduction result of 7.

Common Pitfalls:Attempting to compute 9^19 explicitly; using 9 ≡ −7 (mod 8) and making sign errors; forgetting to add the constant term 6 at the end.

Final Answer:7