2018 · Q42
What is the largest value for n (natural number) such that 6ⁿ divides the product of the first 100 natural numbers?
general-mental-abilityquantitative-aptitude
- A18
- B33
- C44
- D48
Explanation
The exponent of a prime p in 100 factorial is given by Legendre's formula. For 6ⁿ = 2ⁿ·3ⁿ, n = min(v2, v3), where v3 = 33+11+3+1 = 48 and v2 = 50+25+12+6+3+1 = 97. So n = min(97, 48) = 48. Hence the largest natural number n such that 6ⁿ divides 100 factorial is 48.