Question

.Show that square of any positive integer is of the form 4q or 4q + 1 for some integer q.

Answer 1

Answer:

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Answer 2

Step-by-step explanation:

let a be any + ve integer and b=4

then by euclids division algorithm, a=bq+r

a=4q+r

the possible remainders are 0,1,2,3.

then

a=4q+0

a=4q+1

a=4q+2

a=4q+3

case 1

a²=(4q)²

16q² = 4(4q²) where 4q² =q

a² =4q

case 2

a² =( 4q+1)² =16q2+1+8q then we have to take common

a²=4(4q²+2q)+1 (where 4q²+2q =q)

a²= 4q+1

case 3 and case 4 will be little bit similar in this we have to take common and put it as q

so, the square of any +ve integers is in the form of 4q,4q+1

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