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/

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

Question 13 - CBSE Class 10 Sample Paper for 2018 Boards - Part 2
Question 13 - CBSE Class 10 Sample Paper for 2018 Boards - Part 3

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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

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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.