Question
Question: The exponent of 3 in 100! Is...
The exponent of 3 in 100! Is
A
33
B
44
C
48
D
52
Answer
48
Explanation
Solution
Let E(N) denote the exponent of 3 in n. The greatest integer less than 100 divisible by 3 is 99.
We have E(100!) = E(1.2.3.4....99.100)
= E(3.6.9...99) = E[(3.1) (3.2) (3.3) ....(3.33)]
= 33 + E(1.2.3......33)
Now E(1.2.3...33) = E(3.6.9....33)
= E[(3.1) (3.2) (3.3).....(3.11)]
= 11 + E(1.2.3....11)
and E(1.2.3...11) + E(3.6.9) + E[(3.1) (3.2) (3.3)]
3 + E(1.2.3) = 3 + 1 = 4
Thus E(100!) = 33 + 11 + 4 = 48.