1.

Prove that square of any positive integer is of the form 4m or 4m+1​

Answer»

Step-by-step explanation:

APPLYING Euclids division algorithm with a,b,q and r where b=4 (Theorem 1.1)

⇒a=bq+r,0≤r

⇒a=4q+r,0≤r<4

i) When r=0,a=4q

Thus a  

2

=16q  

2

=4(4q  

2

)=4Q where Q=4q  

2

 

ii) When r=1,a=4q+1

⇒a  

2

=(4q+1)  

2

=16q  

2

+6q+1=4q(4q+2)+1=4Q+1

where 4q+2=Q

iii) when r=2,a=4q+2

⇒a  

2

=16q  

2

+16q+4=4(4q  

2

+4q+1)4R

Where a=4q  

2

+4q+1

iv) When r=3,a=4q+3

⇒a  

2

=(4q+3)  

2

=16q  

2

+24q+9=4(4q  

2

+6q+2)+1

=42+1 when 2=4q  

2

+q+1

Thus we can see that the square of any +ve integer is of the form 4Q or 4Q+1 for some integer Q


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