# Problem of the Day

What is the sum of the digits of positive integer ?

(1) The sum of the digits of is an element of the set

(2) for some positive integer .

Reveal Answer### Answer

C. Both statements together are sufficient, but neither statement alone is sufficient.

See the Solution### Solution

[latexpage]

What is the sum of the digits of positive integer $q$?

Statement 1: The sum of the digits of $q$ is an element of the set ${226,313,447,617}$. This is clearly insufficient, as it tells us the answer to the question could be any of these four elements. INSUFFICIENT.

Statement 2: $q=n^3-n$ for some positive integer $n$.Â We can factor this expression as $q=n(n^2-1)=n(n+1)(n-1)=(n-1)n(n+1)$. This expression is the product of three consecutive integers. Because every third integer is a multiple of 3, one of these integers must be a multiple of 3. Thus, $q$ is a multiple of 3. However, this tells us nothing about the sum of the digits of $q$. INSUFFICIENT.

Statements 1&2: Remember that the sum of the digits of a multiple of 3 is also a multiple of 3. Thus, if $q$ is aÂ multiple of 3 and the sum of its digits is either $226,313,447,$ or $617$, it must be $447$ because that is the only multiple of 3 in the set. This is clear because $4+4+7=15$, which is a multiple of 3.

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