Question

Is 4^33 + 3^44 divisible by 5?
Please help ASAP. This is due tomorrow. Thank you very much :)

Answer 1

yes

look at powers of 4
4^1 = 4
4^2 = 16
4^3 = 64
4^4 =256
....
every even indexed power ends in 6, every odd indexed power ends in 4
so 4^33 ends in 4

now look at the powers of 3
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
3^6 = 729
...
the answers end in 3 9 7 1 3 9 7 1 ...
even exponents end in either 9 or 1
if the even exponent is also a multiple of 4, the answer ends in a 1
so 3^44 will end with a 1

so if we add a number ending in 4 to a number ending in 1
we get a number ending in 5 which is divisible by 5