Two consecutive even positive integers, sum of the squares of which is 1060, are

## Concept & Logic This problem maps to forming a quadratic equation based on consecutive even numbers. However, in an exam setting, relying on standard quadratic factoring is slow; understanding unit digits and perfect squares is the primary logical key. Consecutive even integers can be algebraically represented as $x$ and $x + 2$. ## Step-by-Step Solution * **Given:** The sum of the squares of two consecutive even integers is $1060$. * Let the numbers be $x$ and $x + 2$. * Set up the mathematical equation: $$x^2 + (x + 2)^2 = 1060$$ * Expand the binomial: $$x^2 + x^2 + 4x + 4 = 1060$$ $$2x^2 + 4x - 1056 = 0$$ * Simplify by dividing by 2: $$x^2 + 2x - 528 = 0$$ * **Calculation:** Factor the quadratic equation. We need two numbers that multiply to $-528$ and add to $2$. $$(x + 24)(x - 22) = 0$$ * Since the question specifies *positive* integers, $x = -24$ is invalid. Therefore, $x = 22$. * The consecutive even number is $x + 2 = 24$. ## Exam Strategy & Shortcut **Never solve the quadratic equation in the exam.** Use **Option Elimination via Unit Digits**. We need the sum of the squares to end in a $0$ ($1060$). Let's check the unit digits of the squares of the options: * (a) $12$ and $14$: $2^2 + 4^2 = 4 + 16 = 20$. Unit digit is $0$. Sum is $144 + 196 = 340$. Too small. * (b) $20$ and $22$: $0^2 + 2^2 = 0 + 4 = 4$. Unit digit is $4$, not $0$. Reject immediately. * (c) $22$ and $24$: $2^2 + 4^2 = 4 + 16 = 20$. Unit digit is $0$. Let's estimate: $20^2 + 20^2 = 800$. $25^2 + 25^2 = 1250$. The answer must be closely bounded between these. Let's calculate: $484 + 576 = 1060$. This matches perfectly. * (d) $15$ and $18$ are not consecutive even integers. Reject immediately. ## Common Pitfall The main pitfall is wasting 2-3 minutes trying to manually factorize large numbers like $528$ during the quadratic step. Another mistake is forgetting the definition of "consecutive even" and representing the numbers as $x$ and $x+1$. ## Final Answer Therefore, the correct answer is 22 and 24.