Question

Please help!

Find all possible digits P and N satisfying the equation P+P+P+N=NP (bar over NP), where NP (bar over NP) is the two-digit number with N in the tens place and P in the ones place.

The solution must include a convincing argument that you have found all possible answers!

Answer 1

\(\large \begin{align} \color{black}{\normalsize \text{NP (bar over NP) is the two-digit number } \hspace{.33em}\\~\\
\normalsize \text{with N in the tens place and P in the ones place} \hspace{.33em}\\~\\
\normalsize \text{this can be expressed as} \hspace{.33em}\\~\\
\overline{NP} =10N+P\hspace{.33em}\\~\\
\normalsize \text{so} \hspace{.33em}\\~\\
P+P+P+N=\overline{NP} \hspace{.33em}\\~\\
3P+N=10N+P \hspace{.33em}\\~\\
2P=9N \hspace{.33em}\\~\\

P=\dfrac{9}{2}N \hspace{.33em}\\~\\
\normalsize \text{here only even numbers will work for N} \hspace{.33em}\\~\\
\normalsize \text{as odd number cannot be divisible by 2} \hspace{.33em}\\~\\
so ~~N=\{2,4,6,8\cdot \cdot \cdot \cdot \infty \} \hspace{.33em}\\~\\
\normalsize \text{example, when N=2} \hspace{.33em}\\~\\
P=\dfrac{9}{2}N \hspace{.33em}\\~\\
P=\dfrac{9}{2}\times 2 \hspace{.33em}\\~\\
P=9 \hspace{.33em}\\~\\
\normalsize \text{check} \hspace{.33em}\\~\\
9+9+9+2=29 \hspace{.33em}\\~\\
}\end{align}\)

Answer 2

@mathmath333 thank you so much!! but what did it mean when you put "check 9+9+9+2=29"? what does that show/prove?

Answer 3

the assumption was given in the problem that the

where \(\large P\) is the unit's digit
and
\(\large N\) is the ten's digit of \(\large \overline{NP}\)

\(\large \begin{align} \color{black}{P+P+P+N=\overline{NP} \hspace{.33em}\\~\\
\color{blue}{9+9+9}+\color{red}{2}=\color{red}{2}\color{blue}{9}

}\end{align}\)

just gave that to confirm that our approach and the answer is \(\bf correct\)

Answer 4

oh okay, so with that being said, is this a convincing enough argument that we have found all possible answers?

Answer 5

not their is restriction that NP is a two digit number so , u have to find even numbers
for N upto which the NP(bar) is a two digit number only