Finding the highest power of 3 in a factorial product
Question
If 15 × 14 × 13 × ... × 3 × 2 × 1 = 3^m × n where m and n are positive integers, then what is the maximum value of m?
Options
7
6
5
4
Explanation
The problem asks for the maximum power of the prime number 3 that divides 15! (15 factorial) completely.
Use Legendre's Formula to find the exponent of a prime p inside N!: E_p(N!) = \lfloor N/p \rfloor + \lfloor N/p² \rfloor + \lfloor N/p³ \rfloor + \dots
Substitute N = 15 and p = 3: m = \lfloor 15/3 \rfloor + \lfloor 15/9 \rfloor + \lfloor 15/27 \rfloor m = 5 + 1 + 0 = 6.
The maximum power of 3 contained within the product is exactly 6.
Answer: (b).
Question details
Year
2022
Paper
CSAT
Question
Q74
Section
Numerical Ability
Sub-topic
Number System (Highest Power)
Type
Factual single
Difficulty
Medium
Source hint
15x14x...x1 = 3^m * n
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