Is 4^33 + 3^44 divisible by 5?

Please help ASAP. This is due tomorrow. Thank you very much :)

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

To determine if 4^33 + 3^44 is divisible by 5, we need to check if the sum is divisible by 5.

To begin, we can simplify the expression by finding the remainders of the terms when divided by 5.

For 4^33:
The pattern of remainders of the powers of 4 when divided by 5 is: 4, 1, 4, 1, ...
Since 33 is an odd power, the remainder of 4^33 when divided by 5 is also 4.

For 3^44:
The pattern of remainders of the powers of 3 when divided by 5 is: 3, 4, 2, 1, ...
Since 44 is divisible by 4, the remainder of 3^44 when divided by 5 is 1.

Now, we can add the remainders:
4 + 1 = 5

The sum of the remainders is 5, which means that 4^33 + 3^44 is divisible by 5.

Please note that this explanation assumes we are using regular arithmetic rules, and the numbers are integers. If any other non-standard rules or non-integer numbers are involved, this explanation may not be applicable.