Question

show that square of any positive odd integer is of 4m+1 for m
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Answer 1

Step-by-step explanation:

ANSWER

As per Euclid division lemma

if a and b are 2 positive numbers then

a=bq+r,where0<r<b

Let

positive integer be a and b=4

Hence

a=4q+r,where0<r<4

hence,r=0,1,2,3

only for r=1 and 3.the integer a is odd.

then,a=4q+1

Answer 2

Step-by-step explanation:

let x be any positive odd integer

then

x = 2n + 1 for some positive integer n<x

now

x^2 = (2n+1)^2

= 4n^2 + 4n +1

= 4n(n+1) + 1

since n is a positive integer

n(n+1) will also be an integer say m, then

x^2 = 4m + 1

i.e. square of any positive odd integer can be expressed in the form of 4m+1 for some m