Question

Prove that one and only one out of n, (n + 2) and (n + 4) is divisible by 3. Where 'n' is any positive integer.​

Answer 1

We know that any positive integer of the form 3q or, 3q+1 or 3q+2 for some integer q and one and only one of these possibilities can occur.

So, we have following cases:

Case-I: When n=3q

In this case, we have

n=3q, which is divisible by 3

Now, n=3q

n+2=3q+2

n+2 leaves remainder 2 when divided by 3

Again, n=3q

n+4=3q+4=3(q+1)+1

n+4 leaves remainder 1 when divided by 3

n+4 is not divisible by 3.

Thus, n is divisible by 3 but n+2 and n+4 are not divisible by 3.

Case-II: when n=3q+1

In this case, we have

n=3q+1,

n leaves remainder 1 when divided by 3.

n is divisible by 3

Now, n=3q+1

n+2=(3q+1)+2=3(q+1)

n+2 is divisible by 3.

Again, n=3q+1

n+4=3q+1+4=3q+5=3(q+1)+2

n+4 leaves remainder 2 when divided by 3

n+4 is not divisible by 3.

Thus, n+2 is divisible by 3 but n and n+4 are not divisible by 3.

Case-III: When n=3q+2

In this case, we have

n=3q+2

n leaves remainder 2 when divided by 3.

n is not divisible by 3.

Now, n=3q+2

n+2=3q+2+2=3(q+1)+1

n+2 leaves remainder 1 when divided by 3

n+2 is not divisible by 3.

Again, n=3q+2

n+4=3q+2+4=3(q+2)

n+4 is divisible by 3.

Hence, n+4 is divisible by 3 but n and n+2 are not divisible by 3.

Answer 2

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☆Given: -

n,n+2,n+4 is divisible by 3

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☆To prove :-

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

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☆Proof:-

By using Euclid division lemma

we have , a=bq+r , where0 </-r <b

For a =n and b=3 , we have

n=3q+r ,__i)

where q is an integer

and 0 </-r <3 i .e. r=0,1,2.

putting r=0 in i) we get

n=3q

so n is divisible by 3

n+2=3q+2

so n+2is not divisible by 3

n+4=3q+4

so n+4is not divisible by 3

Putting r=1 ii)

n=3q +1

so n is not divisible by 3

n+2=3q+3=3(q+1)

so n+2is divisible by 3.

n+4=3q+ 5.

do n+4is not divisible by 3

Putting r=2 iii)

n=3q+2

so n is not divisible by 3

n+2=3q+4

so n+2is not divisible by 3.

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

so n +4is divisible by 3.

Thus for each value of r such that 0 </-r <3 only one out of n , n+2, n+4 is divisible by 3.

Hope it helps u mate.

Thank you.