Question

use euclids division lemma to prove one and only one out of n, n+2andn+4 is divisible by 3,wher n is any positive integer​

Answer 1

n is an integer. So,it is of the form 3k,3k+1 or 3k+2 where k is an integer.
If n=3k,then
n+1=3k+1,not divisible by 3
n+2=3k+2,not divisible by 3
If n=3k+1(not divisible by 3),then
n+1=3k+2,not divisible by 3
n+2=3k+3=3(k+1), divisible by 3
If n=3k+2(not divisible by 3),then
n+1=3k+3=3(k+1),divisible by 3
n+2=3k+4=3(k+1)+1,not divisible by 3
Hence by seeing all the above cases,we can safely say that:out of n,n+1 and n+2,only one is divisible by 3.

Answer 2

hence, exactly one of the number n, n+2, n+4 is divisible by 3.

hope this helps

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