Question 159
There are many sets of two different positive integers $a$ and $b$, both less than $50$, such that $a^{2}$ and $b^{2}$ end in the same last two digits. For example, $35^{2}=1225$ and $45^{2}=2025$ both end in $25$. What are all possible values for the average of $a$ and $b$?
For the purposes of this problem, single-digit squares are considered to have a leading zero, so for example we consider $2^{2}$ to end with the digits $04$, not $4$.
For the purposes of this problem, single-digit squares are considered to have a leading zero, so for example we consider $2^{2}$ to end with the digits $04$, not $4$.