Question

Cho m,n là 2 số nguyên.Chứng minh rằng nếu 7(m+n)²+2mn chia hết cho 225 thì mn cũng chia hết cho 225

Answer 1

225=15²

=> \(2\left[7\left(m+n\right)^2+2mn\right]⋮15^{^2}\)

\(\Leftrightarrow14\left(m+n\right)^2+4mn⋮15^2\)

\(\Leftrightarrow14\left(m+n\right)^2+\left[\left(m+n\right)^2-\left(m-n\right)^2\right]⋮15^2\)

\(\Leftrightarrow15\left(m+n\right)^2-\left(m-n\right)^2⋮15^2\)

Vì \(15\left(m+n\right)^2⋮15\Rightarrow\left(m-n\right)^2⋮15\)

\(\Rightarrow\left\{{}\begin{matrix}\left(m-n\right)^2⋮3\\\left(m-n\right)^2⋮5\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}m-n⋮3\\m-n⋮5\end{matrix}\right.\)

mà (3,5)=1 => (m-n)\(⋮\)15

=> (m-n)^2\(⋮\)15²

Tương tự 15(m+n)^2\(⋮\)15²

=> mn \(⋮\)225