# Prove that x² - 1 is divisible by a where 8 is odd positive number. Do the solution in stepwise.

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x^2 -1 is divisible by 8, if x is an odd positive integer.

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x^2 -1 is divisible by 8, if x is an odd positive integer.

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Let x be an integer.

x² -1 = (x - 1) (x + 1)

let x be an odd positive integer. x - 1 is an even positive integer or 0 = 2 k (say)

so x + 1 is then an even positive number = 2 k + 2

x² - 1 = 4 k (k + 1)

Now k and k + 1 are consecutive integers. Hence one of them is divisible by 2. If k is even then k+1 is an odd number. If k is an odd number, then k+1 is an even number.

Hence, the product k (k+1) = 2 * [k/2] * (k+1) or k * [ (k+1)/2 ] * 2

thus x² -1 = 8 * (k/2) * (k+1) or 8 * k * [(k+1)/2]

Hence, x² - 1 is divisible by 8.

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you can use the method of proof by induction.

x² - 1 is divisible by 8 if x = 1 .

let x² - 1 be divisible by 8 for some x. Now let us see for the next off number x+2.

(x+2)² - 1

= x² + 4x + 4 - 1

= (x² - 1) + 4 (x + 1)

as x is an odd number, x +1 is an even number. Hence 4 (x+1) is divisible by 8.

Since both terms on RHS are divisible by 8., then the (x+2)² - 1 is divisible by 8.

Hence, proved by mathematical induction method.

x² -1 = (x - 1) (x + 1)

let x be an odd positive integer. x - 1 is an even positive integer or 0 = 2 k (say)

so x + 1 is then an even positive number = 2 k + 2

x² - 1 = 4 k (k + 1)

Now k and k + 1 are consecutive integers. Hence one of them is divisible by 2. If k is even then k+1 is an odd number. If k is an odd number, then k+1 is an even number.

Hence, the product k (k+1) = 2 * [k/2] * (k+1) or k * [ (k+1)/2 ] * 2

thus x² -1 = 8 * (k/2) * (k+1) or 8 * k * [(k+1)/2]

Hence, x² - 1 is divisible by 8.

====================================

you can use the method of proof by induction.

x² - 1 is divisible by 8 if x = 1 .

let x² - 1 be divisible by 8 for some x. Now let us see for the next off number x+2.

(x+2)² - 1

= x² + 4x + 4 - 1

= (x² - 1) + 4 (x + 1)

as x is an odd number, x +1 is an even number. Hence 4 (x+1) is divisible by 8.

Since both terms on RHS are divisible by 8., then the (x+2)² - 1 is divisible by 8.

Hence, proved by mathematical induction method.