The product of two positive integers is 462 . If the first integer is 5 greater than twice the second

Question

The product of two positive integers is 462 . If the first integer is 5 greater than twice the second integer, what is the smaller of the two integers?

Accepted Answer

The correct answer is 14 . Let x represent the first integer and y represent the second integer. If the first integer is 5 greater than twice the second integer, then x = 2 y + 5 . It's given that the product of the two integers is 462 ; therefore x y = 462 . Substituting 2 y + 5 for x in this equation yields ( 2 y + 5 ) ( y ) = 462 , which can be written as 2 y 2 + 5 y = 462 . Subtracting 462 from each side of this equation yields 2 y 2 + 5 y - 462 = 0 . The left-hand side of this equation can be factored by finding two values whose product is 2 ( - 462 ) , or - 924 , and whose sum is 5 . The two values whose product is - 924 and whose sum is 5 are 33 and - 28 . Thus, the equation 2 y 2 + 5 y - 462 = 0 can be rewritten as 2 y 2 - 28 y + 33 y - 462 = 0 , which is equivalent to 2 y ( y - 14 ) + 33 ( y - 14 ) = 0 , or ( 2 y + 33 ) ( y - 14 ) = 0 . By the zero product property, it follows that 2 y + 33 = 0 or y - 14 = 0 . Subtracting 33 from both sides of the equation 2 y + 33 = 0 yields 2 y = - 33 . Dividing both sides of this equation by 2 yields y = - 33 2 . Since y is a positive integer, the value of y isn't - 33 2 . Adding 14 to both sides of the equation y - 14 = 0 yields y = 14 . Substituting 14 for y in the equation x y = 462 yields x ( 14 ) = 462 . Dividing both sides of this equation by 14 yields x = 33 . Therefore, the two integers are 14 and 33 , so the smaller of the two integers is 14 .

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