Question 13
Show that exactly one of the numbers n, n + 2 or n + 4 is divisible by 3.
Theory
As per Euclid’s Division Lemma
If a and b are 2 positive integers, then
a = bq + r
where 0 ≤ r < b
If b = 3,
a = 3q + r
where 0 ≤ r < 3
So, r = 0, 1, 2
∴ Numbers = 3q + 0, 3q + 1, 3q + 2
Let’s assume
n = 3q, 3q + 1, 3q + 2
Now, we check whether n, n + 2, n + 4 is divisible by 3
If n = 3q
n = 3q
Since n can be divided by 3
It is divisible by 3
n + 2 = 3q + 2
Putting q = 1
n + 2 = 3(1) + 2 = 5
Since 5 is not divisible by 3
n + 2 is not divisible by 3
n + 4 = 3q + 4
Putting q = 1
n + 4 = 3(1) + 4 = 7
Since 7 is not divisible by 3
n + 4 is not divisible by 3
If n = 3q + 1
n = 3q + 1
Putting q = 1
n = 3(1) + 1 = 4
Since 4 is not divisible by 3
n is not divisible by 3
n + 2 = 3q + 1 + 2
n + 2 = 3q + 3
n + 2 = 3(q + 1)
Since n + 2 can be divided by 3
It is divisible by 3
n + 4 = 3q + 1 + 4
n + 4 = 3q + 5
Putting q = 1
n + 4 = 3(1) + 5 = 8
Since 8 is not divisible by 3
n + 4 is not divisible by 3
n + 4 = 3q + 1 + 4
n + 4 = 3q + 5
Putting q = 1
n + 4 = 3(1) + 5 = 8
Since 8 is not divisible by 3
n + 4 is not divisible by 3
If n = 3q + 2
n = 3q + 2
Putting q = 1
n = 3(1) + 2 = 5
Since 5 is not divisible by 3
n is not divisible by 3
If n = 3q + 1
n = 3q + 1
Putting q = 1
n = 3(1) + 1 = 4
Since 4 is not divisible by 3
n + 2 is not divisible by 3
n + 4 = 3q + 2 + 4
n + 4 = 3q + 6
n + 4 = 3(q + 2)
Since n + 4 can be divided by 3
It is divisible by 3
We see that in all 3 cases,
Exactly one of the numbers n, n + 2, n + 4 is divisible by 3
Hence proved

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

Hi, it looks like you're using AdBlock :(

Displaying ads are our only source of revenue. To help Teachoo create more content, and view the ad-free version of Teachooo... please purchase Teachoo Black subscription.

Please login to view more pages. It's free :)

Teachoo gives you a better experience when you're logged in. Please login :)

Solve all your doubts with Teachoo Black!

Teachoo answers all your questions if you are a Black user!