Using Euclid's division lemma, show that the square of any positive integer is either of the form 3 m or 3m +1 for some integer m
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Sneha
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Saturday, 28 November 2020 08:38 AM
Let 'a' be any positive integer.
On dividing it by 3 , let 'q' be the quotient and 'r' be the remainder.
Such that ,
a = 3q + r , where r = 0 ,1 , 2
When, r = 0
? a = 3q
When, r = 1
? a = 3q + 1
When, r = 2
? a = 3q + 2
When , a = 3q
On squaring both the sides,
%20%5C%5C%20%7Ba%7D%5E%7B2%7D%20%3D%203%20%5C%5C%20where%20%5C%3A%20m%20%3D%203%20%7Bq%7D%5E%7B2%7D)
When, a = 3q + 1
On squaring both the sides ,
%5E%7B2%7D%20%5C%5C%20%7Ba%7D%5E%7B2%7D%20%3D%209%20%7Bq%7D%5E%7B2%7D%20%2B%202%20%5Ctimes%203q%20%5Ctimes%201%20%2B%20%7B1%7D%5E%7B2%7D%20%5C%5C%20%7Ba%7D%5E%7B2%7D%20%3D%209%20%7Bq%7D%5E%7B2%7D%20%2B%206q%20%2B%201%20%5C%5C%20%7Ba%7D%5E%7B2%7D%20%3D%203(3%20%7Bq%7D%5E%7B2%7D%20%2B%202q)%20%2B%201%20%5C%5C%20%7Ba%7D%5E%7B2%7D%20%3D%203m%20%2B%201%20%5C%5C%20where%20%5C%3A%20m%20%5C%3A%20%3D%203%20%7Bq%7D%5E%7B2%7D%20%2B%202q)
When, a = 3q + 2
On squaring both the sides,
%5E%7B2%7D%20%5C%5C%20%7Ba%7D%5E%7B2%7D%20%3D%203%20%7Bq%7D%5E%7B2%7D%20%2B%202%20%5Ctimes%203q%20%5Ctimes%202%20%2B%20%7B2%7D%5E%7B2%7D%20%5C%5C%20%7Ba%7D%5E%7B2%7D%20%3D%209%20%7Bq%7D%5E%7B2%7D%20%2B%2012q%20%2B%204%20%5C%5C%20%7Ba%7D%5E%7B2%7D%20%3D%20(9%20%7Bq%7D%5E%7B2%7D%20%2B%2012q%20%2B%203)%20%2B%201%20%5C%5C%20%7Ba%7D%5E%7B2%7D%20%3D%203(3%20%7Bq%7D%5E%7B2%7D%20%2B%204q%20%2B%201)%20%2B%201%20%5C%5C%20%7Ba%7D%5E%7B2%7D%20%3D%203m%20%2B%201%20%5C%5C%20where%20%5C%3A%20m%20%5C%3A%20%3D%203%20%7Bq%7D%5E%7B2%7D%20%2B%204q%20%2B%201)
Therefore , the square of any positive integer is either of the form 3m or 3m+1.
On dividing it by 3 , let 'q' be the quotient and 'r' be the remainder.
Such that ,
a = 3q + r , where r = 0 ,1 , 2
When, r = 0
? a = 3q
When, r = 1
? a = 3q + 1
When, r = 2
? a = 3q + 2
When , a = 3q
On squaring both the sides,
When, a = 3q + 1
On squaring both the sides ,
When, a = 3q + 2
On squaring both the sides,
Therefore , the square of any positive integer is either of the form 3m or 3m+1.
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