Question

If n is odd positive integer show that square of n is of the form of 8m +1 for some whole no m

Answer 1

we know that,
a= bq +r equation 1
put b=8 in equation 1
a=8q+r. equation 2
put r=0,1 ,2,3,4,5,6,7 in equation 2
when r=0
a=8q +0
a=8q
squaring both sides
a square =8q*8q
let 8qsquare be m
a square =8m
when r=1
a=8q +1
squaring both sides
a square =8q +1 ka whole square
a square = 8q ka square +1 ka square +2 *8q*1
a square =
a square =64 q square +1 +16q
a square =64q +16q +1
a square=8[8q +2q ]+1
let 8q+2q be m
a square=8m+1

Answer 2

tis is interesting question..beside the proof if u divide a odd no by 2 let the quotient is l the the sum of ist m nautral no will be let m the it will be expressed as 8m+1