Question

Show that one and only one out of n, (n+1) and (n+2) is divisible by 3,where nis
any positive integer.

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Answer 1

$\bold\red{\fbox{\sf{Solution}}}$

• Since n, n+1, n+2 are three consecutive integers then there must be one number divisible by 3 at least. 

• If the remainder at dividing n by 3 is 1, then n+2 must be divisible by 3 and if the remainder at dividing n by 3 is 2, then n+1 must be divisible by 3. Similarly for n+1 and n+2.

• Let n be divisible by 3.

3n+1=3n+31

• Now, n is divisible by 3 but 1 is not. So we get n+1 not divisible by 3. Similarly,n+2 will not be divisible by 3 as well if n is divisible by 3.

3n+2=3n+32

• In the same way, if n+1 is divisible by 3 then n and n+2 can't be divisible by 3. If n+2 is divisible by 3 then n and n+1 cannot be divisible by 3.