Math, asked by Aaaryaa, 4 hours ago

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

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Answered by Anonymous
5

Answer:

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Answered by Anonymous
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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 \orange{n=3q}

In this case, we have

 \red{n=3q}, which is divisible by 3

Now, \blue{n=3q}.

\red{n+2=3q+2}

n+2 leaves remainder 2 when divided by 3

Again,\red{n=3q}

 \red{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

 \green{n=3q+1,}

 \red{n} leaves remainder 1 when divided by 3.

 \red{n} is divisible by 3

Now,  \orange{n=3q+1}

 \orange{n+2=(3q+1)+2=3(q+1)}

n+2 is divisible by 3.

Again,  \red{n=3q+1}

 \orange{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.

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