The difference between two positive integers is $3$. If the sum of their squares is $369$, then the sum of the numbers is

### Concept & Formula You can find the sum of two variables if you know their difference and the sum of their squares by using standard algebraic identities linking $(a+b)^2$, $(a-b)^2$, and $(a^2+b^2)$. $$(a+b)^2 + (a-b)^2 = 2(a^2 + b^2)$$ ### Step-by-Step Solution * **Given:** Let the two positive integers be $x$ and $y$. * Difference: $x - y = 3$ * Sum of squares: $x^2 + y^2 = 369$ * We need to find the sum of the numbers, which is $(x + y)$. * Substitute the given values into the algebraic identity: $$(x+y)^2 + (3)^2 = 2(369)$$ * Square the $3$ and multiply the right side: $$(x+y)^2 + 9 = 738$$ * Isolate $(x+y)^2$: $$(x+y)^2 = 738 - 9$$ $$(x+y)^2 = 729$$ * Take the square root to find the sum: $$x + y = \sqrt{729}$$ $$x + y = 27$$ ### Exam Strategy & Shortcut An alternative approach is to use the formula $(a-b)^2 = a^2 + b^2 - 2ab$ to find $2ab$, and then plug that into $(a+b)^2 = a^2 + b^2 + 2ab$. $3^2 = 369 - 2ab \Rightarrow 2ab = 360$. Then $(a+b)^2 = 369 + 360 = 729 \Rightarrow a+b = 27$. Both algebraic paths are similarly fast, but knowing the direct relationship $(a+b)^2 + (a-b)^2 = 2(a^2 + b^2)$ saves one substitution step and is highly recommended for speed. ### Common Pitfall A common error is to try and guess the numbers by factoring $369$, or guessing numbers that are $3$ units apart (like $12$ and $15$) and manually squaring them ($144 + 225 = 369$). While guessing can sometimes be fast if you are good at mental math, it is unreliable and can waste time if the numbers are not immediately obvious. Stick to the algebraic identity. ### Final Answer **Therefore, the correct answer is 27.**