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The sum of the product and the sum of two positive integers is 454. Find the largest possible value of the product of their sum and their product.

Guest May 14, 2018
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To write this out algebraically, we can assign variables to the two numbers. 

 

First number = x, Second number = y

 

The product of the two numbers is x*y or just xy

 

The sum of the two numbers is (x+y) 

 

The sum of the product and the sum of two positive integers can be expressed as: 

 

xy + x + y, and this is equal to 454, xy + x + y = 454 

 

We can solve for all values of x and y, we get: 

 

xy + x + y = 454

 

We add 1 to both sides so we can factor: 

 

xy + x + y + 1 = 455

 

Then we can factor:

 

x(y+1)+y+1 = 455

 

(y+1)(x+1) = 455 = 5 * 7 * 13

 

Their product is the largest only when x = 90 and y = 4 

 

(90*4)(90+4)=33840

 

Other pairs is not as big, for example, when x = 12 and y = 34

 

(34*12)(34+12)=18768

 

To answer your question, the largest possible value is 33840, 

 

I hope this helped,

 

Gavin

GYanggg  May 14, 2018

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