Ex 1.1 , 4
Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m+ 1 for some integer m. [Hint : Let x be any positive integer then it is of the form 3q, 3q+ 1 or 3q+ 2. Now square each of these and show that they can be rewritten in the form 3m or 3m+ 1.]
As per Euclid’s Division Lemma
If a and b are 2 positive integers, then
a = bq + r
where 0 ≤ r < b
Let positive integer be a
And b = 3
Hence a = 3q + r
where ( 0 ≤ r < 3)
r is an integer greater than or equal to 0 and less than 3
hence r can be either 0 , 1 or 2
Hence, square of any positive number can be expressed of the form 3m or 3m + 1
Hence proved

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.