Remainder of 6¹⁷ + 17⁶ ÷ 7 — Number System Aptitude Question
Remainder of 6¹⁷ + 17⁶ ÷ 7: What is the remainder when 6¹⁷ + 17⁶ is divided by 7? Answer and worked solution on CodeKairo.
- Topic: Number System
- Section: Quantitative Aptitude
- Difficulty: Hard
- Time target: 120 seconds
Question
What is the remainder when 6¹⁷ + 17⁶ is divided by 7?
Options
- 0
- 1
- 5
- 6
Show the answer and the worked solution
Correct answer: A. 0
Worked solution
- 6 ≡ −1 (mod 7), and 17 is odd, so 6¹⁷ ≡ (−1)¹⁷ = −1 ≡ 6 (mod 7).
- 17 ≡ 3 (mod 7). Now 3³ = 27 ≡ 6 ≡ −1, so 3⁶ = (3³)² ≡ 1 (mod 7).
- Sum ≡ 6 + 1 = 7 ≡ 0 (mod 7).
Answer: 0.
The idea
Reduce every base modulo the divisor before touching the exponents, and treat each term separately before adding. A base that reduces to −1 collapses instantly on parity.
Shortcut. Fermat's little theorem gives 17⁶ ≡ 1 (mod 7) with no work, since 7 is prime and does not divide 17; 6 ≡ −1 makes 6¹⁷ ≡ −1, and the two cancel to 0.
Common trap. 6 — the remainder of 6¹⁷ alone, as if 17⁶ added nothing; 17⁶ leaves 1, and 6 + 1 = 7 leaves 0.