Question

N is a multiple of 5
A = N + 1
B = N – 1
Prove, using algebra, that A2 – B2 is always a multiple of 20

Answer 1

Answer:

Proved

$A^2 - B^2 = 20, 40,60,80...$

Step-by-step explanation:

Given

$A = N + 1$

$B = N - 1$

Required

Prove that $A\² - B\²$ is a multiple of 20

First, we need to evaluate $A\² - B\²$

Start by applying difference of two squares

$A\² - B\² = (A + B)(A - B)$

Substitute values for A and B

$A\² - B\²= (N + 1 + N - 1)(N + 1 -(N-1))$

$A\² - B\² = (N + 1 + N - 1)(N + 1 - N + 1)$

Collect Like Terms

$A\² - B\² = (N + N+ 1 - 1)(N - N + 1+ 1)$

$A\² - B\² = (2N)(2)$

$A\² - B\² = 2N*2$

$A\² - B\² = 4N$

Since, N is a multiple of 5, then the possible values of N are:

$N = 5, 10, 15, 20...$

For each of these values, the possible values of $A^2 - B^2$ are

$A^2 - B^2 = 4N: 4 * 5, 4*10,4*15,4*20...$

$A^2 - B^2 = 20, 40,60,80...$

The above shows that $A^2 - B^2$ is a multiple of 20