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 \(3\,\rm{m}\) or \(3\,\rm{m} + 1\) for some integer \(\rm{m.}\)

[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 \(3\,\rm{m}\) or \(3\,\rm{m} + 1.\)]

 Video Solution
Real Numbers
Ex 1.1 | Question 4

Text Solution

To Prove:

The square of any positive integer is either of the form \(3\,\rm{m}\) or \(3\,\rm{m} + 1\) for some integer \(\rm{m}\) (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 “\(\rm{m}\)”) will also be a positive integer. Now, observe carefully that the square of all the positive integers is either of the form \(3\,\rm{m}\) or \(3\,\rm{m} + 1\) for some integer \(\rm{m.}\)

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,

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

Where \(\rm{m}\) is any positive integer.

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

  
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