Problem:
For how many in is the tens digit of odd?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The squares of only two integers in have odd tens digits, and . Since and the tens digit in must be even, it follows that the tens digit in will be odd if and only if the tens digit in is odd. Inductively, we conclude that only numbers in that end in 4 or 6 will have squares with an odd tens digit. There are exactly such numbers.
Since neither nor have odd tens digits, we replace the given set with , and count the of the form
for which has an odd tens digit. Since and is even, it follows that the tens digit of will be odd if and only if the tens digit of is odd. There are two digits, and , for which the tens digit of is odd. Since there are choices for to pair with these two choices for , there are integers in the set whose squares have odd tens digits.