Find the missing digits in this problem about an invoice

[Math100]

Homework Statement

An old and some what illegible invoice shows that $72$ canned hams were purchased for $x67.9y$. Find the missing digits.

Relevant Equations

None.

Consider the missing digits in $x679y$.
Then $72\mid x679y$.
Note that $72=8\cdot 9$.
This means $8\mid x679y$ and $9\mid x679y$.
Now we have
\begin{align*}
&x679y\equiv 0\pmod {9}\\
&\implies (x+6+7+9+y)\equiv 0\pmod {9}\\
&\implies (x+y+22)\equiv 0\pmod {9}\\
&\implies (x+y+4)\equiv 0\pmod {9}.\\
\end{align*}
Thus $x679y=x\cdot 10^{4}+6\cdot 10^{3}+7\cdot 10^{2}+9\cdot 10+y$.
Observe that $x679y\equiv 0\pmod {8}\implies x679y\equiv (4+2+y)\pmod {8}\implies (6+y)\equiv 0\pmod {8}$
because $10^{3}\equiv 0\pmod {8}$ and $10^{4}\equiv 0\pmod {8}$.
Since $(6+y)\equiv 0\pmod {8}\implies y=2$, it follows that $(x+6)\equiv 0\pmod {9}\implies x=3$.
Therefore, the missing digits are $x=3$ and $y=2$.

 

[SammyS]

Homework Statement:: An old and some what illegible invoice shows that $72$ canned hams were purchased for $x67.9y$. Find the missing digits.
Relevant Equations:: None.

Consider the missing digits in $x679y$.
Then $72\mid x679y$.
Note that $72=8\cdot 9$.
This means $8\mid x679y$ and $9\mid x679y$.
Now we have
\begin{align*}
&x679y\equiv 0\pmod {9}\\
&\implies (x+6+7+9+y)\equiv 0\pmod {9}\\
&\implies (x+y+22)\equiv 0\pmod {9}\\
&\implies (x+y+4)\equiv 0\pmod {9}.\\
\end{align*}
Thus $x679y=x\cdot 10^{4}+6\cdot 10^{3}+7\cdot 10^{2}+9\cdot 10+y$.
Observe that $x679y\equiv 0\pmod {8}\implies x679y\equiv (4+2+y)\pmod {8}\implies (6+y)\equiv 0\pmod {8}$
because $10^{3}\equiv 0\pmod {8}$ and $10^{4}\equiv 0\pmod {8}$.
Since $(6+y)\equiv 0\pmod {8}\implies y=2$, it follows that $(x+6)\equiv 0\pmod {9}\implies x=3$.
Therefore, the missing digits are $x=3$ and $y=2$.

Correct.

I would determine $y$ first. We must have that $8$ divides $79y$, the right most 3 digits, That will give you that $y=2$ .

After that, it's almost trivial to find that $x=3$ .