Isaiah’s Favorite Trick of Factoring

My search for math wizards still continue. This time, another wizard named Isaiah James Maling would like to share another problem. Isaiah started loving math when he was in high school at Ramon Magsaysay High School Manila. He has struggles early in high school life dealing with his math subject but it didn’t stop him in his determination to be a great mathematician. After pure hard work and self-study, Isaiah then became one of the most formidable math wizards of his school. In fact, he didn’t even lose a division contest for MTAP starting 1st year until his 4th year (2003-2007) . He has also carried his team to 2 championship sectoral level competition and 1 regional 1st runner-up. He continued his excellence in mathematics in his tertiary level by joining National and Inter-Collegiate Math Quiz Show and he emerged victorious in the 12th and 13th MSP Search for the NCR College Math Wizard (2011 and 2012) when he was declared champion individually and the National MTAP for Tertiary Level in 2011 by being a champion in both Team and Individual Category.

Problem:

If a and b are positive integers for which ab–3a+4b=137, what is the minimal possible value of  | a – b | ?

Solution:

By applying Simon’s Favorite Factoring Trick and note that if we subtract 12 from both sides, then the left side can be factored. Thus,

ab – 3a + 4b – 12 = 125

(a + 4)(b – 3) = 125

Since a and b are positive integers, then (a + 4) and (b – 3) must be a pair of factors of 125 =  so (a + 4) and (b – 3) must be among (1,125) , (5,25) , (25,5) , (125,1)

Thus, (a,b) must be among

(-3,128) , (1,28) , (21,8) , (121,4).

 Ruling out the first solution on account of the negative value for a, we must find the minimal value of  | a – b | among the remaining three is |21 – 8| = 13