Question

show that 1 and only 1 out of n, n+2, n+4 is divisible by 3 where n is any positive integer

Answer 1

"show that 1 and only 1 out of n, n+2, n+4 "
Could you draw that out? I don't know what you mean by that.

Answer 2

The question means that for any value of n one of the functions above will be divisible by 3. For example if n = 1. \(n=1\) \(n+2=3\) \(n+4=5\)
So for that example n+2 is divisible by 3. Does that make sense?

Try to look at the difference between each function to get you started

Answer 3

Start off with assuming there exists at least 2 out of them which are divisible by 3 and consider the difference between them, you would arrive at a contradiction.

Answer 4

I was thinking about using mathematical induction.

Answer 5

Using mathematical induction will be right @Azteck .. But there is an easy method too.

Answer 6

Yeah I was thinking of the easy method, but I can't grasp it. I will see what you come up with mathslover.

Answer 7

You have to wait for that... - 1 minute pls

Answer 8

Yeah I will be on another tab helping someone else.

Answer 9

Well by Euclid's division lemma we can write : \(n = 3q + r\) where q is quotient and r is remainder.

Answer 10

We have \(0\le r < 3\)

Answer 11

So we have 3 cases : r = 0 , r = 1 and r = 2

\(\textbf{Case 1:} \space n = 3q + 0\) ---- \(r = 0 \)
\(n = 3q\)
\(n+2 = 3q +2\)

Answer 12

So it means when \(n+2\) is divided by 3 it gives remainder as 2 . Hence, n+2 is not divisible by 3 ------------ (1)

Answer 13

Similarly : n + 4 = 3q + 4

that is : n +2 = 3q + 3 + 1

or : n + 4 = 3(q+1) + 1
or : n + 4 = 3m + 1 where m is an integer.

thus n+4 is also NOT divisible by 3 .

Answer 14

Hence when "n" is divisible by 3 , then it is NOT DIVISIBLE BY n+2 and n+4

Answer 15

Similarly when n+2 is divisible by 3 then n and n+4 will not be divisible by 3.

and when n+4 is divisible by 3 then n and n+2 will not be divisible by 3.

Answer 16

Got it @queenbhavya , what are your thoughts @Azteck

Answer 17

AH I see. Great work mathslover. Well thought out.

Answer 18

:) @queenbhavya did you get what I did?

Answer 19

thnkuuuuuu

Answer 20

Actually considering the differences between any 2 of them is prolly the easiest way to do it, but if you are more familiar with this kind of proof then go for it.

Answer 21

Best of Luck queenbhavya and : \[
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Answer 22

if n+4 is divisible by three , then so is n+1


n, n+1, n+2, are three consecutive integers
so one of these will be divisible by three