How-many-integers-are-there-such-that-0-n-720-and-n-2-1-mod-720-

Question Number 211222 by RojaTaniya last updated on 31/Aug/24

H o w m a n y i n t e g e r s a r e t h e r e

s u c h t h a t, 0 ⩽ n ⩽ 720 a n d

n^{2} \equiv 1 (m o d) 720 ?

Answered by mahdipoor last updated on 31/Aug/24

\Rightarrow n^{2} = 1 + 720 m

\Rightarrow n^{2} \equiv 1 (m o d 2) \Rightarrow n = 2 k + 1

{(2 k + 1)}^{2} = 720 m + 1 \Rightarrow 4 k^{2} + 4 k + 1 = 720 m + 1 \Rightarrow

k^{2} + k = 180 m

\Rightarrow k^{2} + k \equiv 0 m o d 9 \Rightarrow k = 9 j o r 9 j + 8

\Rightarrow k^{2} + k \equiv 0 m o d 5 \Rightarrow k = 5 i o r 5 i + 4

k = 9 j = 5 i \Rightarrow k = 45 t

k = 9 j = 5 i + 4 \Rightarrow k = 45 t + 9

k = 9 j + 8 = 5 i \Rightarrow k = 45 t + 35

k = 9 j + 8 = 5 i + 4 \Rightarrow k = 45 t + 44

\Rightarrow k^{2} + k \equiv 0 m o d 4 \Rightarrow k = 4 i o r 4 i + 3

k = 4 i = 45 t \Rightarrow k = 180 r

k = 4 i = 45 t + 9 \Rightarrow k = 180 r + 144

k = 4 i = 45 t + 35 \Rightarrow k = 180 r + 80

k = 4 i = 45 t + 44 \Rightarrow k = 180 r + 44

k = 4 i + 3 = 45 t \Rightarrow k = 180 r + 135

k = 4 i + 3 = 45 t + 9 \Rightarrow k = 180 r + 99

k = 4 i + 3 = 45 t + 35 \Rightarrow k = 180 r + 35

k = 4 i + 3 = 45 t + 44 \Rightarrow k = 180 r + 179

\Rightarrow\Rightarrow k = 180 r + f_{n}

\Rightarrow n = 2 k + 1 = 360 r + (2 f_{n} + 1) = 360 r + d_{n}

d_{n} = 1, 71, 89, 161, 199, 271, 289, 359 r \in Z

\Rightarrow 0 ⩽ n ⩽ 720 \Rightarrow

n = 1, 71, 89, 161, 199, 271, 289, 359

361, 431, 449, 521, 559, 631, 649, 719