Question

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

Answer 1

Answer:

Let 'a' be any positive integer.

b=4

by Euclid's division lemma,

a=bq+r

a²=(bq+r)²-------1.

r=0,1,2,3

from 1.

for r=0,

a²=(4q+0)²

a²=16q²

a²=4(4q²)

=4q, where q=4q².

for r=1,

a²=(4q+1)²

a²=16q²+1+8q

a²=4(4q²+2q)+1

a²=4q+1, where q=4q²+2q.

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Answer 2

Answer:

Let `a´ be positive integer

b = 4

by Euclid's division lemma,

a = bq +r

a square = (bq + r) square

r = 0,1,2,3

from 1

for r = 0

a square = (4q + 0) square

a square = 16q square

a square = 4(4q) square

= 4q, where q= 4q square

for r = 1

a square = (4q + 1) square

a square = 16q square + 1 +8q

a square = 4(4q square + 2q) + 1

a square = 4q + 1

Where, q = 4q square + 2q

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