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Question Number 211222 by RojaTaniya last updated on 31/Aug/24
How many integers are there such that, 0≤n≤720 and n^2 ≡1(mod)720?
$${How}\:{many}\:{integers}\:{are}\:{there} \\ $$$$\:\:\:{such}\:{that},\:\:\mathrm{0}\leqslant{n}\leqslant\mathrm{720}\:{and}\: \\ $$$$\:\:\:{n}^{\mathrm{2}} \equiv\mathrm{1}\left({mod}\right)\mathrm{720}? \\ $$
Answered by mahdipoor last updated on 31/Aug/24
⇒n^2 =1+720m ⇒n^2 ≡1 (mod 2) ⇒ n=2k+1 (2k+1)^2 =720m+1⇒4k^2 +4k+1=720m+1⇒ k^2 +k=180m ⇒k^2 +k≡0 mod 9 ⇒k=9j or 9j+8 ⇒k^2 +k≡0 mod 5 ⇒k=5i or 5i+4 k=9j=5i ⇒ k=45t k=9j=5i+4 ⇒ k=45t+9 k=9j+8=5i ⇒ k=45t+35 k=9j+8=5i+4⇒ k=45t+44 ⇒k^2 +k≡0 mod 4 ⇒k=4i or 4i+3 k=4i=45t ⇒ k=180r k=4i=45t+9 ⇒ k=180r+144 k=4i=45t+35 ⇒ k=180r+80 k=4i=45t+44 ⇒ k=180r+44 k=4i+3=45t ⇒ k=180r+135 k=4i+3=45t+9 ⇒ k=180r+99 k=4i+3=45t+35⇒ k=180r+35 k=4i+3=45t+44⇒ k=180r+179 ⇒⇒k=180r+f_n ⇒n=2k+1=360r+(2f_n +1)=360r+d_n d_n =1,71,89,161,199,271,289,359 r∈Z ⇒ 0≤n≤720 ⇒ n=1,71,89,161,199,271,289,359 361,431,449,521,559,631,649,719
$$\Rightarrow{n}^{\mathrm{2}} =\mathrm{1}+\mathrm{720}{m}\:\:\: \\ $$$$\Rightarrow{n}^{\mathrm{2}} \equiv\mathrm{1}\:\:\left({mod}\:\mathrm{2}\right)\:\Rightarrow\:{n}=\mathrm{2}{k}+\mathrm{1} \\ $$$$\left(\mathrm{2}{k}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{720}{m}+\mathrm{1}\Rightarrow\mathrm{4}{k}^{\mathrm{2}} +\mathrm{4}{k}+\mathrm{1}=\mathrm{720}{m}+\mathrm{1}\Rightarrow \\ $$$${k}^{\mathrm{2}} +{k}=\mathrm{180}{m} \\ $$$$\Rightarrow{k}^{\mathrm{2}} +{k}\equiv\mathrm{0}\:\:\:{mod}\:\mathrm{9}\:\Rightarrow{k}=\mathrm{9}{j}\:{or}\:\mathrm{9}{j}+\mathrm{8} \\ $$$$\Rightarrow{k}^{\mathrm{2}} +{k}\equiv\mathrm{0}\:\:\:{mod}\:\mathrm{5}\:\Rightarrow{k}=\mathrm{5}{i}\:{or}\:\mathrm{5}{i}+\mathrm{4} \\ $$$${k}=\mathrm{9}{j}=\mathrm{5}{i}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:{k}=\mathrm{45}{t} \\ $$$${k}=\mathrm{9}{j}=\mathrm{5}{i}+\mathrm{4}\:\:\:\:\:\:\:\Rightarrow\:\:{k}=\mathrm{45}{t}+\mathrm{9} \\ $$$${k}=\mathrm{9}{j}+\mathrm{8}=\mathrm{5}{i}\:\:\:\:\:\:\:\Rightarrow\:\:{k}=\mathrm{45}{t}+\mathrm{35}\: \\ $$$${k}=\mathrm{9}{j}+\mathrm{8}=\mathrm{5}{i}+\mathrm{4}\Rightarrow\:\:{k}=\mathrm{45}{t}+\mathrm{44} \\ $$$$\Rightarrow{k}^{\mathrm{2}} +{k}\equiv\mathrm{0}\:\:\:{mod}\:\mathrm{4}\:\Rightarrow{k}=\mathrm{4}{i}\:\:{or}\:\:\mathrm{4}{i}+\mathrm{3} \\ $$$${k}=\mathrm{4}{i}=\mathrm{45}{t}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:{k}=\mathrm{180}{r} \\ $$$${k}=\mathrm{4}{i}=\mathrm{45}{t}+\mathrm{9}\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:{k}=\mathrm{180}{r}+\mathrm{144} \\ $$$${k}=\mathrm{4}{i}=\mathrm{45}{t}+\mathrm{35}\:\:\:\:\:\:\:\Rightarrow\:\:{k}=\mathrm{180}{r}+\mathrm{80}\:\: \\ $$$${k}=\mathrm{4}{i}=\mathrm{45}{t}+\mathrm{44}\:\:\:\:\:\:\:\Rightarrow\:\:{k}=\mathrm{180}{r}+\mathrm{44} \\ $$$${k}=\mathrm{4}{i}+\mathrm{3}=\mathrm{45}{t}\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:{k}=\mathrm{180}{r}+\mathrm{135} \\ $$$${k}=\mathrm{4}{i}+\mathrm{3}=\mathrm{45}{t}+\mathrm{9}\:\:\:\Rightarrow\:\:{k}=\mathrm{180}{r}+\mathrm{99}\:\: \\ $$$${k}=\mathrm{4}{i}+\mathrm{3}=\mathrm{45}{t}+\mathrm{35}\Rightarrow\:\:{k}=\mathrm{180}{r}+\mathrm{35}\: \\ $$$${k}=\mathrm{4}{i}+\mathrm{3}=\mathrm{45}{t}+\mathrm{44}\Rightarrow\:\:{k}=\mathrm{180}{r}+\mathrm{179} \\ $$$$\Rightarrow\Rightarrow{k}=\mathrm{180}{r}+{f}_{{n}} \\ $$$$\Rightarrow{n}=\mathrm{2}{k}+\mathrm{1}=\mathrm{360}{r}+\left(\mathrm{2}{f}_{{n}} +\mathrm{1}\right)=\mathrm{360}{r}+{d}_{{n}} \\ $$$${d}_{{n}} =\mathrm{1},\mathrm{71},\mathrm{89},\mathrm{161},\mathrm{199},\mathrm{271},\mathrm{289},\mathrm{359}\:\:\:\:\:{r}\in{Z} \\ $$$$\Rightarrow\:\mathrm{0}\leqslant{n}\leqslant\mathrm{720}\:\Rightarrow \\ $$$${n}=\mathrm{1},\mathrm{71},\mathrm{89},\mathrm{161},\mathrm{199},\mathrm{271},\mathrm{289},\mathrm{359} \\ $$$$\:\:\:\:\:\:\:\mathrm{361},\mathrm{431},\mathrm{449},\mathrm{521},\mathrm{559},\mathrm{631},\mathrm{649},\mathrm{719} \\ $$

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