Question

show that any positive integer is of the form 4q+1 or 4q+3, where 'q' is some integer

Answer 1

a=bq+r. 0=r<b
the possible values of r are0,1,2,3
let take p as an intiger
let take value of r=1
a=4p+1. (squaring on both sides)
a2=(4p+1)2
a2=16p2+1+8p
a2=4(4p2+2p)+1
a2=4q+1
as same u can do other parts

Answer 2

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let a be any positive integer

then

b= 4

a= bq+r

0≤r<b

0≤r<4

r= 0,1,2,3

case 1.

r=0

a= bq+r

4q+0

4q

case 2.

r=1

a= 4q+1

6q+1

case3.

r=2

a=4q+2

case 4.

r=3

a=4q+3





hence from above it is proved that any positive integer is of the form 4q, 4q+1,4q+2,4q+3

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