# 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 .

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

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### 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.