Question

[I lay my love on you]

A number m becomes a square number when 14 is subtracted from it.The sum of m and 19 is another square number.Find the value of m

Answer 1

[Nguyễn Tất Đạt]

We have: \(m-14=h^2\) (h² is a square number)

And \(m+19=k^2\)(k² is a square number)

\(\Rightarrow m+19-\left(m-14\right)=k^2-h^2\)\(\Leftrightarrow33=\left(k-h\right)\left(k+h\right)\)

So (k-h) and (k+h) is a pair of estimate numbers of 33

Estimate numbers of 33 are 1;3;11;33;-1;-3;-11;-33

+) If \(k-h=1\Rightarrow k+h=33\)(because k-h < k+h)

As \(k-h=1>0\Rightarrow k>h\)\(\Rightarrow h=\frac{33-1}{2}=16\Rightarrow k=17\)

\(\Rightarrow\hept{\begin{cases}m+19=17^2\\m-14=16^2\end{cases}}\Leftrightarrow\hept{\begin{cases}m+19=289\\m-14=256\end{cases}}\Rightarrow m+19+m-14=545\)

\(\Leftrightarrow2m+5=545\Leftrightarrow2m=540\Leftrightarrow m=270\)

+) If \(k-h=3\Rightarrow k+h=11\Rightarrow\hept{\begin{cases}k=7\\h=4\end{cases}}\Rightarrow\hept{\begin{cases}m+19=49\\m-14=16\end{cases}}\Rightarrow2m+5=65\Leftrightarrow m=30\)

+) If \(k-h=-33\Rightarrow k+h=-1\Rightarrow\hept{\begin{cases}k=-17\\h=16\end{cases}}\Rightarrow\hept{\begin{cases}m+19=289\\m-14=256\end{cases}}\Rightarrow m=270\)

+) If \(k-h=-11\Rightarrow k+h=-3\Rightarrow\hept{\begin{cases}k=-7\\h=4\end{cases}}\Rightarrow\hept{\begin{cases}m+19=49\\m-14=16\end{cases}}\Rightarrow m=30.\)

So \(m\in\left\{270;30\right\}.\)