Number Theory Problem 1

Factoring gives us $(x+y)(x-y)=37$

Since $x$ and $y$ are positive and $37$ is prime, we have:

$x+y=37$

$x-y=1$

Therefore $x=19$ and $y=18$

difference of 2 consecutive number's squares are in sequence 1,3,5,7,9..... so, let x & y are 2 positive consecutive numbers. so, x=y+1, x^2-y^2=37, (x-y)(x+y)=37, so 2y+1=37, y=18, so, x+2y=55

$\begin{aligned} x^2-y^2&=37 \\ (x+y)(x-y)&=37 \\ \end{aligned}$

$x$ and $y$ are $2$ positive integers.

$\begin{aligned} x+y&=37 \\ x-y&=1 \\ \implies x&=19 \\ y&=18 \\ \implies x+2y&=19+2(18)=19+36=\boxed{55} \\ \end{aligned}$