Prove-that-any-integer-can-be-expressed-as-in-the-form-of-4k-or4k-1-or-4k-2-

Question Number 44059 by paro123 last updated on 20/Sep/18

Prove that any integer can be expressed

as in the form of 4 k or 4 k \underline{+} 1 or 4 k \underline{+} 2 .

Answered by kunal1234523 last updated on 21/Sep/18

a n y i n t e g e r p c a n b e e x p r e s s e d i n t h e f o r m

p = 4 q \pm r

h e r e, r = 0, 1, 2, 3

c a s e 1

r = 0

p = 4 q \pm 0 = 4 q

s i m i l a r l y f o r

r = 1 p = 4 q \pm 1

r = 2 p = 4 q \pm 2

f o r r = 3

p = 4 q \pm 3

p = 4 q \pm 4 \mp 1

p = 4 (q \pm 1) \mp 1

p = 4 m \mp 1 = 4 m \pm 1, m = q \pm 1

t h i s i s s a m e f o r r = 1

s o p = 4 q, 4 q \pm 1, 4 q \pm 2