Math-question-for-anyone-who-has-a-bonus-code-solution-Question-Find-the-smallest-numbers-that-are-divisible-by-15-remain-9-and-divide-by-25-remain-19-and-divide-by-18-remain-12-remain-and-divide

Question Number 168026 by mathlove last updated on 01/Apr/22

$$ \\ $$Math question for anyone who has a bonus code solution.
Question. Find the smallest numbers that are divisible by 15 remain 9 and divide by 25 remain 19 and divide by 18 remain 12 remain and divide by 36 remain 30 remain and divide above 40 remain 34 remain
Whoever solves it on the sheet, put his photo in the comment again.

Commented by mr W last updated on 01/Apr/22

$$\mathrm{1794} \\ $$

Answered by mr W last updated on 01/Apr/22

$${say}\:{the}\:{number}\:{is}\:{N},\:{it}'{s}\:{to}\:{see}\:{it}\:{must} \\ $$$${be}\:{a}\:{multiple}\:{of}\:\mathrm{6},\:{i}.{e}. \\ $$$${N}=\mathrm{6}{n} \\ $$$$ \\ $$$$\mathrm{6}{n}=\mathrm{15}{a}+\mathrm{9}\:\:\:\:…\:\left(\mathrm{1}\right) \\ $$$$\Rightarrow\mathrm{2}{n}−\mathrm{5}{a}=\mathrm{3}\: \\ $$$$\Rightarrow{n}=\mathrm{5}{h}−\mathrm{1} \\ $$$$ \\ $$$$\mathrm{6}\left(\mathrm{5}{h}−\mathrm{1}\right)=\mathrm{25}{b}+\mathrm{19}\:\:\:…\left(\mathrm{2}\right) \\ $$$$\Rightarrow\mathrm{6}{h}−\mathrm{5}{b}=\mathrm{5} \\ $$$$\Rightarrow{h}=\mathrm{5}{k} \\ $$$$ \\ $$$$\mathrm{6}×\left(\mathrm{5}×\mathrm{5}{k}−\mathrm{1}\right)=\mathrm{18}{c}+\mathrm{12}\:\:\:…\left(\mathrm{3}\right) \\ $$$$\Rightarrow\mathrm{5}×\mathrm{5}{k}=\mathrm{3}\left({c}+\mathrm{1}\right) \\ $$$$\Rightarrow{k}=\mathrm{3}{l} \\ $$$$ \\ $$$$\mathrm{6}×\left(\mathrm{5}×\mathrm{5}×\mathrm{3}{l}−\mathrm{1}\right)=\mathrm{36}{d}+\mathrm{30}\:…\left(\mathrm{4}\right) \\ $$$$\Rightarrow\mathrm{5}×\mathrm{5}{l}=\mathrm{2}\left({d}+\mathrm{1}\right) \\ $$$$\Rightarrow{l}=\mathrm{2}{m} \\ $$$$ \\ $$$$\mathrm{6}×\left(\mathrm{5}×\mathrm{5}×\mathrm{3}×\mathrm{2}{m}−\mathrm{1}\right)=\mathrm{40}{e}+\mathrm{34}\:\:\:\:…\left(\mathrm{5}\right) \\ $$$$\mathrm{3}×\mathrm{5}×\mathrm{3}{m}=\mathrm{2}\left({e}+\mathrm{1}\right) \\ $$$$\Rightarrow{m}=\mathrm{2}{s} \\ $$$$ \\ $$$$\Rightarrow{the}\:{number}\:{is} \\ $$$${N}=\mathrm{6}×\left(\mathrm{5}×\mathrm{5}×\mathrm{3}×\mathrm{2}×\mathrm{2}{s}−\mathrm{1}\right)=\mathrm{6}\left(\mathrm{300}{s}−\mathrm{1}\right) \\ $$$${the}\:{smallest}\:{is}\:{N}_{{min}} =\mathrm{6}×\mathrm{299}=\mathrm{1794} \\ $$

Commented by mathlove last updated on 01/Apr/22

$${thanks}\:{mr}\:{W} \\ $$

Commented by Tawa11 last updated on 01/Apr/22

$$\mathrm{Great}\:\mathrm{sir} \\ $$

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