If n is a positive two-digit integer, how many different values of n allow n^3 - n to be a multiple of 12?

Question

Can someone please help me with this?

Sanjana Patil · Accepted Answer

Total two-digit integers are 90.
n^3-n= n(n^2-1)= (n-1)*(n)*(n+1) -> 3 consecutive integers
For n=10 product of 3 consecutive integer is not multiple of 12,For n=14 product is not multiple of 12.
These number will form arithmetic progression starting from 10 and ending in 98. Series will be 10,14,18,....90,94,98
Number of terms98= 10+(n-1) *4n=23
There are 23 two-digits number which are not multiple of 12.
Total multiple of 12 which are two - digit integers are 90 - 23 = 67&shy;