Question

show that square of an odd positive integer is expressed in the form 8m+1, where m is a positive integer

Answer 1

by euclids divison lemma for any 2 positive integers a and b there exits a unique interger q and r such that a=bq+r  ,, 0 is < or equal to r is greater than b

 

the possible values of r may be0,1,2,3,4,5,6,7.

if r=0 then a=8q+0

=a2=(8q)2

a2=64q2

  =8(8q2)

here 8q2 is m

if r=1 then a=8m+1

a2=(8q +1)2

  =64q2 + 16q+1

  = 8(8q2+2q)+1

here m= 8q2+2q


hope it helps you!!