Question

show the square of odd positive integer is of the form 8q + 1 for some positive integer q​

Answer 1

Step-by-step explanation:

Let 'a' be the any positive integer.

Then, b = 8 .

Using Euclid's division lemma :-

0 ≤ r < b => 0 ≤ r < 8 .

$\therefore$ 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 .

Answer 2

NCERT Class X

Mathematics - Exemplar Problems

Chapter _QUESTION PAPER SET II

Answer:

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

✔✔ Proved✔✔

Step-by-step explanation:

Given Probelm:

Show the square of odd positive integer is of the form 8q + 1 for some positive integer q

Solution:​

⇒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.

According to your question:

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 .