M18-33

Official Solution:


If \(m\) and \(n\) are non-negative integers, what is the units digit of \(5^m + 6^n\) ?

Note that \(m\) and \(n\) are non-negative integers, which means each of them can be 0, 1, 2, and so on. Also, observe that the units digit of 5 raised to a positive integer power is always 5, and the units digit of 6 raised to a positive integer power is always 6. So, if we knew that \(m\) and \(n\) are positive integers, instead of non-negative integers, the answer would be 1, because ...5 + ...6 = ...1

(1) \(m + n > 2\)

Both \(m\) and \(n\) can be positive, giving the units digit of the sum as 1. However, one of them can be 0, and another greater than 2, and in this case, the units digit of \(5^m + 6^n\) will be either 6 (if \(n = 0\)) or 7 (if \(m = 0\)). Not sufficient.

(2) \(mn > 0\)

The above implies that \(m\) and \(n\) are either both negative or both positive. However, since it is given that \(m\) and \(n\) are non-negative, they cannot be both negative, and thus both must be positive. Therefore, the units digit of \(5^m + 6^n\) is 1. Sufficient.


Answer: B