Remainder of a factorial sum — Aptitude Question
Remainder of a factorial sum: What is the remainder when 1! + 2! + 3! + … + 100! is divided by 24? Answer and worked solution on CodeKairo.
- Topic: Number System
- Section: Quantitative Aptitude
- Difficulty: Medium
- Time target: 60 seconds
Question
What is the remainder when 1! + 2! + 3! + … + 100! is divided by 24?
Options
- 6
- 9
- 15
- 21
Show the answer and the worked solution
Correct answer: B. 9
Worked solution
- 4! = 24, and every factorial after it contains 4! as a factor, so 4! + 5! + … + 100! is divisible by 24.
- Only 1! + 2! + 3! = 1 + 2 + 6 = 9 is left.
- 9 < 24, so the remainder is 9.
Answer: 9.
The idea
In a factorial sum, everything from the divisor's own factorial onwards vanishes. Add the handful of terms below it and reduce — the sum is never large.
Shortcut. Stop at 3!: every term from 4! = 24 on is a multiple of 24. The sum is odd (1 plus even terms), which rules out 6 at a glance, and 1 + 2 + 6 = 9.
Common trap. 15 — it is 24 − 9, the amount needed to reach the next multiple of 24, confused with the remainder itself.