Question

Show that the square of any positive odd integer is in the form of 8m+1 for some integer m

Answer 1

let a be any positive integer
b=4(we can take 4)
by Euclid algorithm
a=bq+r
the common factors are 0 1 2 3 4
for 0,
4q(even)
4q+1,it is not find so it is odd
4q+2=q=4/2=2 (even)
4q+3=odd
4q+4=even

Answer 2

$\huge \bf{ \red{hey}}$

$\huge{ \mathfrak{here \: is \: answer}}$

let a be any positive integer

then

b=8

0≤r<b

0≤r<8

r=0,1,2, 3,4,5,6,7

case 1.

r=0

a=bq+r

8q+0

(8q)^2

=> 64q^2

8(8q^2)

= let 2q^2 be m

8m

case 2.
r=1
a=bq+r

(8q+1)^2

(8q)^2+2*8q*1+(1)^2

64q^2+16q+1

8(8q^2+2q)+1.

let 8q^2+2q be. m

8m+1

case 3.

r=2

(8q+2)^2

(8q)^2+2*8q*2+(2)^2

64q^2+32q+4

8(8q^2+4q)+4

let 8q^2+4q be m

8m+4

case4.

r=3
(8q+3)^2

(8q)^2+2*8q*3+(3)^2

64q^2+48q+9

64q^2+48q+8+1

8(8q^2+6q+1)+1

let 8q^2+6q+1 be m

8m+1

case 5.

r=4

(8q+4)^2

64q^2+64q+16
8(8q^2+8q+2)

let 8q^2+8q+2 be m

8m

case 6

r=5

(8q+5)^2

64q^2+80q+25

64q^2+80q+24+1

8(8q^2+10q+3)+1

let 8q^2+10q+3 be m

8m+1

case 7

r=6

(8q+6)^2

64q^2+96q+36

64q^2+96q+32+4

8(8q^2+12q+4)+4

let 8q^2+12q+4 be m..

8m+4

case8.

r=7

(8q+7)^2

64q^2+112q+49

64q^2+112q+48+1

8(8q^2+14q+6)+1

let 8q^2+14q+6 be m

8m+1

from above it is proved.

$\huge \boxed{ \boxed{ \pink{hope \: it \: helps}}}$

$\huge{ \green{thanks}}$