Question

# Given N is a positive integer less than 31, how many values can n take if (n + 1) is a factor of n!

CAT 2021

A
B
C
D
##### 20
Solution

Correct option is

(A)

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"