17. Sarah chooses two numbers $a$ and $b$ from the set $\{1,2,3, \ldots, 26\}$. The product $a b$ is equal to the sum of the remaining 24 numbers.
What is the difference between $a$ and $b$ ?
Solution
006

The sum of the numbers in the set is $\frac{1}{2} \times 26 \times 27=351$. Numbers $a$ and $b$ will satisfy the equation $351-a-b=a b$.

Therefore $352=a b+a+b+1=(a+1)(b+1)$. We now search for a factor pair $(a+1, b+1)$ of 352 with $a, b \leq 26$ and $a \leq b$. The only such pair $(a+1, b+1)$ is $(16,22)$. Therefore $a$ and $b$ are 15 and 21 respectively and their difference is 6 .