Find the exponent of 80 in 200!.

Answer

49

Explanation

`80=2^(4)xx5`
To find the exponent of 80 in 200!, we find the exponent of 2 and 5. Exponent of 2 is
`[(200)/(2)]+[(200)/(2^(2))]+[(200)/(2^(3))]+[(200)/(2^(4))]+[(200)/(2^(5))]+[(200)/(2^(6))]+[(200)/(2^(7))]`
`=100+50+25+12+6+3+1`
=197
Exponent of 5 of
`[(200)/(5)]+[(200)/(5^(2))]+[(200)/(5^(3))]=40+8+1=49`
Now, exponent of 16 in 200! is `[197//4]=49`. Hence, exponent of 80 is 49.