Question

use division algorithm to show that the square of any positive integer is of the form 3p or 3p+1.​

Answer 1

Answer:

Use euclids division lemma to show that the square of any poistive integer is of the form 3p. ,3p+1... Use euclids division lemma to show that the cube of any positive integer is of the form 3p or 3 +1 ... Use euclids division lemma to show that the spare of any positive integer is of the form of 3p , 3p... answer

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Answer 2

$Let ′a′ be any positive integer and b=3,</p><p>Then, by division algorithm a=3q+r,</p><p>for some integer q⩾0 and 0⩽r⩽3.</p><p>So, a=3q,3q+1,3q+2.</p><p>Then, the square of positive integer,</p><p></p><p>a=3q</p><p>a2=(3q)2</p><p>     =9q2</p><p>     =3(3q2)</p><p>     =3p</p><p>(Where p=3q2) </p><p></p><p></p><p>a=3q+1</p><p></p><p>a2=(3q+1)2</p><p></p><p>      =9q2+6q+1</p><p></p><p>      =3(3q2+2q)+1</p><p></p><p>      =3p+1</p><p></p><p>(Where p=3q2+2q)</p><p></p><p></p><p></p><p>a=3q+2</p><p></p><p>a2=(3q+2)2</p><p></p><p>      =9q2+12q+4</p><p></p><p>      =3(3q2+4q+1)+1</p><p></p><p>      =3p+1</p><p></p><p>(Where p=3q2+4q+1)</p><p></p><p>Since p is some positive integer.</p><p>∴ The square of any positive integer is of the form 3p or 3p+1.</p><p></p><p>$

Step-by-step explanation:

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