2010/022) What is the largest number of 0's that can occur at the end of 1^n+2^n +3^+4^n

for every positive integer n

1^n + 2^n + 3^n + 4^n

n = 1 gives 10 and hence one zero
n =2 gives 41 so zero
n =3 gives 100 so 2 zeros

and it cannot end with > 2 zeros as it can be proved in steps as below

for it to end with k zeroes it need to be divisible by 10^k that is 2^k and 5^k. If we can show that is it not divisible by 8 then we are through.

for n > = 3, 2^n and 4^n are divisible by 8

now 1^n =1 for all n
3^n = 1 mod 8 for even n and 3 mod 8 for odd n

So 1^n + 3^n = 2 mod 8 for n even and 4 mod 8 odd n

So not divisible by 8 for any n and hence it cannot have >2 zeros

so ans is 2(for n = 3)