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Question -

Prove that one and only one out of n, n + 2and n + 4 is divisible by 3, where n is any positive integer.



Answer -

Solution:

According to Euclid’s division Lemma,

Let the positive integer = n

And b=3

n =3q+r, where q is the quotient and r is theremainder

0

Therefore, n may be in the form of 3q, 3q+1,3q+2

When n=3q

n+2=3q+2

n+4=3q+4

Here n is only divisible by 3

When n = 3q+1

n+2=3q=3

n+4=3q+5

Here only n+2 is divisible by 3

When n=3q+2

n+2=3q+4

n+4=3q+2+4=3q+6

Here only n+4 is divisible by 3

So, we can conclude that one and only one outof n, n + 2 and n + 4 is divisible by 3.

 

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