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Class 10 > Learn Mathematics > Prove that x2 -- x is divisible by 2 for all positive...
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Prove that x2 – x is divisible by 2 for all positive integer x.

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The expression x^2-x always gives even number for any integer , so that it always divisible by 2
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The integer =z( -4,-3,-2,-1,0,1,2,3,4). Now positive integer is 1,2,3,4........the question is , X2 – X where X is a positive integer ,always divisible by 2 . If we put 1,2,3 in place of X we find 12 -1 =0 ,22 – 2 =2 , 32 -3 = 6 .so we can say “ X will be 2,3,4,5,……..(X...
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 The integer =z( -4,-3,-2,-1,0,1,2,3,4). Now positive integer is  1,2,3,4........the question is  , X2 – X    where X  is a positive integer ,always divisible by 2 .  If we put  1,2,3  in place of X we find   12 -1  =0  ,22 – 2 =2  ,  32 -3 = 6 .so  we can  say “  X   will be 2,3,4,5,……..(X will not be  1).”

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We prove this by mathematical induction method If x=1 we get x2-x=0 Statement is true for x=1 Assume that statement is true for x=n, i.e., n2-n=2p for some p. Now consider, x=n+1 (n+1)2-(n+1)=n2+1+2n-n-1=n2+n= 2p+n+n=2p+2=(2p+1) Statement is true for x=n+1 Therefore statement is true for x is a any positive...
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We prove this by mathematical induction method

If x=1 we get x2-x=0

Statement is true for x=1

Assume that statement is true for x=n, i.e., n2-n=2p for some p.

Now consider, x=n+1

(n+1)2-(n+1)=n2+1+2n-n-1=n2+n= 2p+n+n=2p+2=(2p+1)

Statement is true for x=n+1

Therefore statement is true for x is a any positive integers

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