Problem 4 · Two numbers from the list 1 to 50
The add-one trick: factoring $(a+1)(b+1) = 1276$.
Integer answer, at most 4 digitsWe list all natural numbers from 1 to 50. We remove two numbers from the list, in such a way that the product of these two numbers equals the sum of the numbers remaining in the list. Which are these two numbers? (As your answer, write the two numbers consecutively, starting with the smaller one. For example, if the numbers were 7 and 83, you would write 783.)
Reasoned solution
Key idea: the sum $1 + 2 + \cdots + 50 = \dfrac{50 \cdot 51}{2} = 1275$. If we remove $a$ and $b$, the condition is $ab = 1275 - a - b$.
Move everything to one side and add $1$ so we can factor:
Factor: $1276 = 2^{2} \cdot 11 \cdot 29$. We need two factors between $2$ and $51$ (since $1 \le a < b \le 50$). The only valid decomposition is
Check: $28 \cdot 43 = 1204 = 1275 - 28 - 43$ ✓. The other divisor pairs ($22 \cdot 58$, $11 \cdot 116$...) involve a factor larger than $51$.