Ex.1.1 Q4 Real Numbers Solution - NCERT Maths Class 10

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Question

Use Euclid’s division lemma to show that the square of any positive integer is either of the form \(3\,m \) or \(3\,m + 1\) 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 \(3\,m\) or \(3\,m + 1.\)]

 Video Solution
Real Numbers
Ex 1.1 | Question 4

Text Solution

To Prove:

The square of any positive integer is either of the form \(3\,m \) or \(3\,m + 1\) for some integer \(m\) (using the Euclid’s division lemma).

Reasoning:

Suppose that there is a positive integer ‘\(a\)’. By Euclid’s lemma, we know that for positive integers \(a\) and \(b,\) there exist unique integers \(q\) and \( r,\) such that \(a=bq+r,\,0\le r\lt b\)

If we keep the value of \(b = 3,\) then \(0 ≤ r < 3\) i.e. \(r = 0\) or \(1\) or \(2\) but it can’t be \(3\) because \(r\) is smaller than \(3.\) So, the possible values for \(a = 3q\) or \(3q + 1 \) or \(3q + 2.\) Now, find the square of all the possible values of \(a.\) If \(q\) is any positive integer then its square (let’s call it as “\(m\)”) will also be a positive integer. Now, observe carefully that the square of all the positive integers is either of the form \(3\,m \) or \(3\,m + 1\) for some integer \(m. \)

Steps:

Let \(“a” \) be any positive integer and \(b=3.\)

Then, \(a=3q+r\) for some integer \(q\ge 0\) and \(r = 0, 1 , 2\) because \(0\le r <3.\)

Therefore,

\[\begin{align} a&=3q \;\text{ or }\;3q+1\;\text{ or }\;3q+2\text{ or }\;\\ {{a}^{2}}&\!=\!{{\left( 3q \right)}^{2}}\;\!\text{ or }\!\;{{\left(3q+1 \right)}^{2}}\;\!\!\text{ or }\!\!\;{{\left( 3q{ }+{ }2 \right)}^{2}} \\ {{a}^{2}}&\!\!=\!\!3{{(3q^{2})}}\;\!\!\text { or }\;\!\!\!(9{{q}^{2}}\!\!+\!\!6q\!+\!1)\;\!\!\\&\text{ or }\!\!\!\;(9{{q}^{2}}\!+\!12q\!+\!{{4}^{}}) \\{{a}^{2}}&=3\left( 3{{q}^{2}} \right)\;\text{ or }3\left( 3{{q}^{2}}+2q \right)+1\\&\qquad \text{ or }\;3\left( 3{{q}^{2}}+{ }4q+1 \right)+1 \\&=m \; \text{ or } \;3 m +1 \\\end{align}\]

Where \(m \) is any positive integer.

Hence it can be said that the square of any positive integer is either of the form \(\begin{align}{3\,m \;\rm{or}\; 3\,m + 1.} \end{align}\)

  
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