Question

prove that x square - x is divisible by 2 for all positive integers x​

Answer 1

First we are going to check whether x² - x is even or not if x is an odd number.

Let x = 2k - 1, as  x is odd.

Such that,

⇒ x² - x

⇒ x(x - 1)

⇒ (2k - 1)(2k - 1 - 1)

⇒ (2k - 1)(2k - 2)

⇒ 2(2k - 1)(k - 1)

Here, it seems that x² - x is a multiple of 2 if x is an odd number.

Hence proved that x² - x is even if x is odd.

Now let x be even.

In the factorization of x² - x, it's x(x - 1), here it means that x² - x is also even if x is even. Because x² - x is product of x and x - 1, and we considered that x is even, and also we know that the product of a number and an even number is always even. Here x - 1 is odd.

∴  x² - x  is divisible by 2 for all 'integers' x.

(x can be a negative integer too.)

   

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Answer 2

Step-by-step explanation:

To prove : x

2

−x is divisible by 2∀x∈N

x

2

−x=x(x−1)

x and (x−1) are consecutive numbers

So, if x=even (x−1)=odd.

if x=odd (x−1)=even

⇒x(x−1) is always even

⇒x(x−1) is always divisible by 2 for all positive integer.