Question

show that the square of an odd positive integer is of the form 8m+1 where m is some whole number ​

Answer 1

Answer:

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Answer 2

Answer:

Let a be any postive integer and b=4

Then, by Euclid's algorithm a=4q+r for some integer 0≤q and 0 ≤r<4

Thus r = 0,1,2,3

So, a is an odd integer so a=4q+1 or 4q+3

Case 1 : When a =4q+1

Squaring both the sides,we have a²=(4q+1)²

a²=16q²+1+8q

a²=8(2q²+q)+1

a²=8m+1 where m=2q²+q

Case 2: When a=4q+3

Squaring both the sides,we have

a²=(4q+3)²

a²=16q²+9+24q

a²=16q²+24q+8+1

a²=8(2q²+3q+1)+1

a²=8m+1 where m=2q²+3q+1

Hence, a is form 8m+1 for some integer m.