Question

show that he square of the odd integers on the form of4q+1 for same integer q​

Answer 1

Answer:

a=bq+r

(4q+1)²=4q²+2(4q)(1)+1²

=16q²+8q+1

=4(4q²+2q) +1

We have shown that 4q+1 is a positive odd integer on the same integer q

Answer 2

Step-by-step explanation:

let a be any any positive odd integer and let b = 4

using Euclid division lemma

there exist two unique integer q and r which satisfy a=bq+ r ; 0 <r<b

it implies a = 4q+r ; 0<r<4

therefore r is also integer and possible values of r are 1,2,3

case 1: when r = 1 in a= 4q+ r

a=4q+ 1

squaring both sides

a²= (4q+1)²

using Id. (a+b)² = a²+ b²+ 2ab

a²= (4q)² + 1² + 2(4q)(1)

a²= 16q² +1 + 8q

a² = 4( 4q²+2q)+1

a² = 4q + 1 , where 4q²+ 2q