The exponent of 12 in 100! Is

Options

A. 24
B. 46
C. 47
D. 48

Answer

D

Explanation

We have,
`12=2^(2)xx3=2xx2xx3and100!""=2^(a).3^(b).5^(c)....`
Now,
`a=E_(2)(100!)`
`a=[(100)/(2)]+[(100)/(2^(2))]+[(100)/(2^(4))]+[(100)/(2^(5))]+[(100)/(2^(6))]`
`impliesa=50+25+12+6+3+1=97`
and,
`b=E_(3)(100!)`
`impliesb=[(100)/(3)]+[(100)/(3^(2))]+[(100)/(3^(3))]+[(100)/(3^(4))]=33+11+3+1=48`
`:.100!""2^(97)xx3^(48)xx5^(c)xx..........`
`implies100!""(2^(2)xx3)^(48)xx2xx5^(c)xx.............`
`implies100!""=12^(48)xx2xx5^(c)xx............`
Hence, `E_(12)(100!)=48`