Question

Q23. Show that one and only one of n, n + 2 and
n + 4 is divisible by 3. (CBSE 2008 F)
[NCERT Exemplar]
pls Tell​

Answer 1

Answer:

We know that any positive integer is 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

⇒n+2 is not divisible 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 not 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 is not divisible by 3

Again, n=3q+2

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

⇒n+4 is divisible by 3

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

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