sanju09 wrote:If n = 5^50, how many different remainders result when n, n + 7, n + 11, n + 18, n + 21, n + 29, and n + 32 are each divided by 11?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
Nice question!
The first thing we should recognize is that n/11 and (n+11)/11 will have the same remainder.
Example: Let n = 20, which means n+11=31
20/11 gives us a remainder of 9
31/11 gives us a remainder of 9
Notice that 11 divides into 20 one time, with remainder 9, and 11 divides into 31 two times, with remainder 9.
Since 31 is 11 greater than 20, we can see that 11 divides into 31 one more than 11 divides into 20 and we are left with the same remainder.
Similarly, n+7, n+18 and n+29 will all have the same remainder when divided by 11
Here's why:
Let's say that (n+7)/11 gives us a quotient of q and remainder of k.
This means that 11 divides into n+7 q times with k left over.
Since n+18 is 11 more than n+7, we know that (n+18)/11 will give us a quotient of q+1 and remainder of k.
Similarly, (n+29)/11 will give us a quotient of q+2 and remainder of k.
Finally, using similar logic, we can see that n+21 and n+32 will all have the same remainder when divided by 11
So, the answer is
A
Cheers,
Brent
