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Question Number 125670 by mathocean1 last updated on 12/Dec/20

N<10200 , N has five digits. N≡22[23] and N≡5[17]. Determinate the integer N.

$${N}<\mathrm{10200}\:,\:{N}\:{has}\:{five}\:{digits}. \\ $$ $${N}\equiv\mathrm{22}\left[\mathrm{23}\right]\:{and}\:{N}\equiv\mathrm{5}\left[\mathrm{17}\right]. \\ $$ $${Determinate}\:{the}\:{integer}\:{N}. \\ $$

Answered by floor(10²Eta[1]) last updated on 12/Dec/20

10000≤N<10200 N≡22(mod23)⇒N=23a+22, a∈Z N≡5(mod17)⇒23a+22≡6a+5≡5(mod17) ⇒6a≡0(mod17)⇒a≡0(mod17) ⇒a=17b∴N=391b+22, b∈Z 10000≤391b+22<10200 9978≤391b<10178 26≤b≤26⇒b=26 ⇒N=391.26+22 N=10188

$$\mathrm{10000}\leqslant\mathrm{N}<\mathrm{10200} \\ $$ $$\mathrm{N}\equiv\mathrm{22}\left(\mathrm{mod23}\right)\Rightarrow\mathrm{N}=\mathrm{23a}+\mathrm{22},\:\mathrm{a}\in\mathbb{Z} \\ $$ $$\mathrm{N}\equiv\mathrm{5}\left(\mathrm{mod17}\right)\Rightarrow\mathrm{23a}+\mathrm{22}\equiv\mathrm{6a}+\mathrm{5}\equiv\mathrm{5}\left(\mathrm{mod17}\right) \\ $$ $$\Rightarrow\mathrm{6a}\equiv\mathrm{0}\left(\mathrm{mod17}\right)\Rightarrow\mathrm{a}\equiv\mathrm{0}\left(\mathrm{mod17}\right) \\ $$ $$\Rightarrow\mathrm{a}=\mathrm{17b}\therefore\mathrm{N}=\mathrm{391b}+\mathrm{22},\:\mathrm{b}\in\mathbb{Z} \\ $$ $$\mathrm{10000}\leqslant\mathrm{391b}+\mathrm{22}<\mathrm{10200} \\ $$ $$\mathrm{9978}\leqslant\mathrm{391b}<\mathrm{10178} \\ $$ $$\mathrm{26}\leqslant\mathrm{b}\leqslant\mathrm{26}\Rightarrow\mathrm{b}=\mathrm{26} \\ $$ $$\Rightarrow\mathrm{N}=\mathrm{391}.\mathrm{26}+\mathrm{22} \\ $$ $$\mathrm{N}=\mathrm{10188} \\ $$

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