Question

The product of two positive integers plus their sum is 103. The integers are relatively prime, and each is less than 20. What is the sum of the two integers?

Hint: Use "Simon's Favorite Factoring Trick"

Answer 1

19

Explanation:

Let our numbers be $a$ and $b$ with $a>b.$ Then $ab+a+b=103$. With Simon's Favorite Factoring Trick in mind, we add $1$ to both sides and get $ab+a+b+1 = 104$, which factors as $(a+1)(b+1)=104$. We consider pairs $(a+1, b+1)$ of factors of $104$: $(104,1), (52,2), (26,4), (13,8)$. Since $a<20$, we can rule out the first 3 pairs, which gives $a=12$ and $b=7$, so $a+b=\boxed{19}$.

Answer 2

`xy+x+y=103\implies xy+x+y+1=104\implies (x+1)(y+1)=104`

104 has only one pair of divisors less than 20,


`104=8\cdot13=(x+1)(y+1)`

which means
`x+y=7+12=19`.