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If $a$ and $b$ are positive integers for which $ab - 3a + 4b = 131$, what is the minimal possible value of $|a - b|$?

 Jun 3, 2024
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ab -3a + 4b  = 131        take the product of the coefficients on a,b.....add this product to  both sides

 

ab - 3a + 4b + (4 * - 3)  = 131 + (4 * -3)

 

ab - 3a + 4b - 12  =  119            factor the left side

 

(a + 4) ( b - 3)  = 119

 

Factors of  119  are    1   7    17     119

 

Using   17 and  7

 

( 13 + 4) ( 10 - 3)   →  l a - b l  =  l13 - 10 l =  3

 

cool cool cool

 Jun 3, 2024

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