Question

Show that the swaure of any positive integer is of the form 4m or 4m+1 where m is nay integre\

Answer 1

Step-by-step explanation:

let a be positive integer

a=4m

squaring on both sides

asq=4msq

as =16msq

asq=16msq

asq=4m

case2

a=4m+1

squaring on both sides

asq= 4msq+1sq + (4m) (1)

asq= 16msq + 1 + 4m

a= 4(4m+1)

squaring of any positive integer is in the form of 4m,4m+1

Answer 2

Answer:

4m; 4m+1

Step-by-step explanation:

let us assume that any positive integer is a form of a and b =4

by euclid's Lemma a + bq + r = 0 where b is equal to 4 and R is 1,2,3

show that a + 4 q+r=0

so possible values of a are

a =4q+0

a=4q+1

a=4q+2

a=4q+3

squaring both sides

a²=(4q) ²= 16q²=4(4q²)=4m where m=4q²

a²=(4q+1)²=16q²+1²+2×4q×1=4(4q²+2q)+1=4m m=4q²+2q

a²=(4q+2)²=16q²+4+16q=4(4q²+1+4q)=4m m=4q²+1+4q

a²=(4q+3)²=16q²+9+24q=4(4q²+4+6q)+1

4m+1. m=4q²+4+6q

so that positive every positive integer is form of 4m and 4m+1