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The number 24! is divisible by...

The number `24!` is divisible by

A

`6^(24)`

B

`24^(6)`

C

`12^(12)`

D

`48^(5)`

Text Solution

Verified by Experts

The correct Answer is:
B
We have,
`E_(2)(24!)=[(24)/(2)]+[(24)/(2^(2))]+[(24)/(2^(3))]+[(24)/(2^(4))]`
`=12+6+3+1=22`
and, `E_(3)(24!)=[(24)/(3)]+[(24)/(3^(2))]=8+2=10`
`:.24!""=2^(22)xx3^(10)=(2^(7))xx3^(10)xx2`
`=(2^(3)xx3)^(7)xx3^(3)xx2=(24)^(7)xx3^(3)xx2`
Clearly, 24! is divisible by `24^(6)`.
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