Question

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

Answer 1

Maybe this is what I can do:

All of them can NOT be divisible by 3, because if you add a number divisible by 3 and a number not divisible by 3, we get a sum not divisible by 3. Thus if any of these is divisible by 3, then only one of them.

Answer 2

sorry , but i am unable to understand the second line .

Answer 3

nevermind, we can use induction. Want to?

Answer 4

http://www.algebra.com/algebra/homework/real-numbers/real-numbers.faq.question.242615.html

Answer 5

can't we do this through Euclid's division lemma ?

Answer 6

Honestly, I don't know that. Thanks for giving me something to look at:)

Answer 7

in ur ans adding 1 to both sides , is that part of the induction rule or we just do it as a compulsory thing ?

Answer 8

cause' in 10 std we haven't got induction.

Answer 9

At the final step when they had k+3=3m, they add 1,
and they obtain k+4=3m+1.
(Where m and k are natural numbers).

So they are shoing the final case, where k+3 (or just k) is divisible by 3,
then the rest are not divisible by 3.

A precisely, at that step thaqt you asked, they showed that then k+4 is NOT divisible by 3, because it is same thing as 3m+1 which is not divisible by 3.

Answer 10

And Euclid's division lemma is something I have always been mentally doing, but never know it is called this way. LOL
Happens to me sometimes:D

Answer 11

can u then please explain me this que through Euclids division lemma, cause' this is in our course ?

Answer 12

I am not very sure about doing it. Don't want to get it wrong-:(
I am guessing you would start from supposing that
\(\large\color{black}{ \displaystyle n=3k+r }\)
where m>k

Answer 13

i mean n>k

Answer 14

Sorry about that.

Answer 15

You can tag other people who are better than me though, like this @sonali46 .
Good luck!

Answer 16

TO tag, just put the @ key in front of the person's username whom you desire to tag.

Answer 17

where should i write this ?

Answer 18

We know that any positive integer is of the form 3q or 3q + 1 or 3q + 2 for some integer q & one and only one of these
possibilities can occur
Case I : When n = 3q
In this case, we have,
n=3q, which is divisible by 3
n=3q
= adding 2 on both sides
n + 2 = 3q + 2
n + 2 leaves a remainder 2 when divided by 3
Therefore, n + 2 is not divisible by 3
n = 3q
n + 4 = 3q + 4 = 3(q + 1) + 1
n + 4 leaves a 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 a reaminder 1 when divided by 3
n is not divisible by 3
n = 3q + 1
n + 2 = (3q + 1) + 2 = 3(q + 1)
n + 2 is divisible by 3
n = 3q + 1
n + 4 = 3q + 1 + 4 = 3q + 5 = 3(q + 1) + 2
n + 4 leaves a 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
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 divsible by 3
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