Question

prove that the square of an odd positive integer is of the form 8q +1 from some integer q.

Answer 1

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Answer 2

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Let 'a' be the any positive integer.

Then, b = 8 .

Using Euclid's division lemma :-

0 ≤ r < b => 0 ≤ r < 8 .

•°• The possible values of r is 0, 1, 2, 3, 4, 5, 6, 7.

▶ Question said Square of odd positive integer , then r = 1, 3, 5, 7 .


→ Taking r = 1 .

a = bm + r .

= (8q + 1)² .

= 64m² + 16m + 1

= 8( 8m²+ 2m ) + 1 .

= 8q + 1 . [ Where q = 8m² + 2m ]

→ Taking r = 3 .

a = bq + r .

= (8q + 3)² .

= 64m² + 48m + 9 = 64m² + 48m + 8 + 1 .

= 8( 8m²+ 6m + 1 ) + 1 .

= 8q + 1 . [ Where q = 8m² + 6m + 1 ]


→ Taking r = 5 .

a = (8q + 5)² .

= 64m² + 80m + 25 = 64m² + 80m + 24 + 1 .

= 8( 8m²+ 10m + 3 ) + 1 .

= 8q + 1 . [ Where q = 8m² + 10m + 3 ]


→ Taking r = 7 .

a = ( 8q + 7 )² .

= 64m² + 112m + 49 = 64m² + 112m + 48 + 1 .

= 8( 8m²+ 14m + 6 ) + 1 .

= 8q + 1 . [ Where q = 8m² + 14m + 6 ] .


Hence, the square of any odd positive integer is of the form 8q + 1 .

✓✓ Proved ✓✓



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