Let @x denote the sum of the digits of the positive integer x. For example, @8 = 8 and $@123$ = 1 + 2 + 3 = 6. For how many two-digit values of x is @(@x) = 3 ?

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Official Explanation

Official

@(@x) = 3 denotes that the sum of the digits of the sum of the digits of x = 3.

This can occur in one of two ways:

1) The sum of the digits of x is 3.

2) The sum of the digits of x is 12.

Two-digit numbers which sum to 3:

12, 21

30

(3 in total)…

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What this question tests

What it tests

Your ability to evaluate and compose functions, including unusual defined operations.

Common trap

Applying function operations in the wrong order or ignoring domain restrictions.

Let @x denote the sum of the digits of the positive integer x. For example, @8 = 8 and @123@123@123 = 1 + 2 + 3 = 6. For how many two-digit values of x is @(@x) = 3 ?