2021 AMC 10 A Fall
Complete problem set with solutions and individual problem pages
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?(2021 AMC Fall 10A, Question #8)
- A.
- B.
- C.
- D.
- E.
Solution 1:
Note that the number . By the problem statement, From this we see that must be divisible by 9 . This only happens when . Then, . Thus, there is only cuddly number, which is 89 .
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Solution 2:
If the tens digit is and the ones digit is then the number is so we have the equation . We can guess and check after narrowing the possible cuddly numbers down to , and 99 . (We can narrow it down to these by just thinking about how 's value affects 's value and then check all the possiblities.) Checking all of these we get that there is only (B) 1 2-digit cuddly number, and it is 89 .
