### Show that exactly one of the numbers n, n + 2 or n + 4 is divisible by 3.

This is a question of CBSE Sample Paper - Class 10 - 2017/18.

You can download the question paper here https://www.teachoo.com/cbse/sample-papers/

Last updated at Sept. 14, 2018 by Teachoo

This is a question of CBSE Sample Paper - Class 10 - 2017/18.

You can download the question paper here https://www.teachoo.com/cbse/sample-papers/

Transcript

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

CBSE Class 10 Sample Paper for 2018 Boards

Paper Summary

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Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13 You are here

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20

Question 21

Question 22

Question 23

Question 24

Question 25

Question 26

Question 27

Question 28

Question 29

Question 30

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.