Problem:
A two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?
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Solution:
The value of a two-digit cuddly number is . Therefore , so . It follows that 9 divides . If and , then , which is a contradiction. Thus either or . Furthermore, implies . Therefore the only possible value for is 9 . Then , so , and the only two-digit cuddly number is 89 .
The problems on this page are the property of the MAA's American Mathematics Competitions