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

A
B
C
D
##### 25 and 4
Solution

Correct option is

(C)

According to ques , the number is a multiple of $6^{10}$

if n! is a multiple of $6^{10}$ , it has to be a multiple of $3^{10}$

the smallest factorial should be 24! , so , when n= 24, 25 or 26 , n! will be multiple of $6^{10}$ (not $6^{11}$)

similarly ,we have to find n! such that it is a multiple of $2^{21}$

but not $2^{24}$ . when n = 24 , 25 , 26 , 27 , n! will be multiple of $2^{21}$ but not $2^{24}$

hence the smallest n from 24 , 25 , 0r 26 , will be 24 .

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