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Problem of the Week
Problem D and Solution
More Factors to Consider
Problem
Two integers are multiplied together. The smaller integer is then added to the
product, resulting in the sum 391. Find the four integer pairs that satisfy the
conditions of the problem.
Solution
Let x represent the smaller integer.
Let y represent the larger integer. We know that y > x.
The product of the two integers is xy. To this product we add the smaller
integer x. The resulting expression is xy + x. We want to find all possible pairs
of integers satisfying the equation xy + x = 391.
Factoring the left side of the equation, we obtain x(y + 1) = 391. We want the
product of two integers to be 391. Either both integers are positive or both
integers are negative. The factors of 391 are {±1, ±17, ±23, ±391}.
Then 391 = (−391) × (−1) = (−23) × (−17) = 1 × 391 = 17 × 23.
The first integer in each product is the smaller of the two integers and is
therefore the value of x. The second integer in each pair is larger and
corresponds to y + 1. So, the value of y is 1 less than the second integer in each
of the products.
Therefore, the ordered pairs satisfying xy + x = 391 are (x, y) = (−391, −2),
(x, y) = (−23, −18), (x, y) = (1, 390), and (x, y) = (17, 22).
We can check the validity of each pair by substituting into the expression
xy + x to confirm that the value is 391.
When x = −391 and y = −2,
xy + x = (−391)(−2) + (−391) = 782 − 391 = 391.
When x = −23 and y = −18, xy + x = (−23)(−18) + (−23) = 414 − 23 = 391.
When x = 1 and y = 390, xy + x = (1)(390) + (1) = 390 + 1 = 391.
When x = 17 and y = 22, xy + x = (17)(22) + (17) = 374 + 17 = 391.