Question

# A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions

CAT 2021

Correct option is

**Explanatory Answer**

According to ques , the number is a multiple of

if n! is a multiple of , it has to be a multiple of

the smallest factorial should be 24! , so , when n= 24, 25 or 26 , n! will be multiple of (not )

similarly ,we have to find n! such that it is a multiple of

but not . when n = 24 , 25 , 26 , 27 , n! will be multiple of but nothence the smallest n from 24 , 25 , 0r 26 , will be 24 .

hence C is the correct option , "24 , 3"