Remainder of 2⁵¹ ÷ 7 — Number System Aptitude Question
Remainder of 2⁵¹ ÷ 7: What is the remainder when 2⁵¹ is divided by 7? Answer and worked solution on CodeKairo.
- Topic: Number System
- Section: Quantitative Aptitude
- Difficulty: Medium
- Time target: 60 seconds
Question
What is the remainder when 2⁵¹ is divided by 7?
Options
- 1
- 2
- 4
- 6
Show the answer and the worked solution
Correct answer: A. 1
Worked solution
- 2³ = 8 ≡ 1 (mod 7).
- 51 = 3 × 17, so 2⁵¹ = (2³)¹⁷ ≡ 1¹⁷ = 1 (mod 7).
Answer: 1.
The idea
Look for the smallest exponent that brings the base to 1 modulo the divisor, then reduce the given exponent by that cycle length. This beats Fermat's theorem for small numbers because the cycle is often shorter than p − 1.
Shortcut. Powers of 2 leave 2, 4, 1 on division by 7, repeating every 3. The digits of 51 sum to 6, so 51 is a multiple of 3 and 2⁵¹ lands on the last value of the cycle, 1.
Common trap. 2 — reading 51 mod 3 = 0 as the first position of the cycle 2, 4, 1; a zero remainder means the last position.