Question

A certain number gives a remainder 5 on dividing by 7 and a remainder 7 on dividing by 15. Find the smallest such number.

Answer 1

Dear student
$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{required}\mathrm{number}\mathrm{be}\mathrm{z}. \\ \mathrm{Then}\mathrm{According}\mathrm{to}\mathrm{question}, \\ \mathrm{z}=7\mathrm{x}+5\mathrm{and}\mathrm{z}=15\mathrm{y}+7 \\ \Rightarrow 7\mathrm{x}+5=15\mathrm{y}+7 \\ \Rightarrow 7\mathrm{x}=15\mathrm{y}+2..\left(1\right) \\ \mathrm{Here}\mathrm{x}=15\mathrm{n}-2,\mathrm{y}=7\mathrm{n}-2\mathrm{satsfies}\mathrm{the}\mathrm{eq}\left(1\right) \\ \mathrm{Putting}\mathrm{n}=1,\mathrm{we}\mathrm{get} \\ \mathrm{x}=15-2=13\mathrm{and}\mathrm{y}=7-2=5 \\ \mathrm{So},\mathrm{z}=15\times 5+7=82 \\ \mathrm{Thus},\mathrm{the}\mathrm{smallest}\mathrm{number}\mathrm{which}\mathrm{leaves}\mathrm{remainder}5\mathrm{and}7\mathrm{when}\mathrm{divided}\mathrm{by} \\ 7\mathrm{and}15\mathrm{respectively}\mathrm{is}82. \end{gathered}$$
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