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Among grandfather's papers a bill was found:

$$72 \text{ turkeys \$}-67.9- $$

The first and last digit of the number that obviously represented the total price of those fowls are replaced here by blanks, for they have faded and are now illegible.

What are the two faded digits and what was the price of one turkey?

Now this is how far I got, I realized that if convert the $ to cent I get the a integer. Now if this integer is divisible by 72 then it is divisible by 9 and 8. Now this is where I get stuck, they take 79- and state that this has to be divisible by 8 since 1000 is divisible by 8. But I know several digits that are less then 1000 that is divisible by 9 so why would it have to be 8 ? I'll post the full answer later.

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  • $\begingroup$ Because for 9 you would need to get the sum of the digits but you have 2 unknowns so there is no possible way. With 8 you just need the the last 3 digits to be divisible by 8 so you can solve for one unknown digit. $\endgroup$ – Ziad Fakhoury Jun 20 '16 at 20:40
  • $\begingroup$ Not following your objection. $x6000$ is divisible by $8$ because $1000$ is. Thus $x679y$ is divisible by $8$ if and only if $79y$ is. $\endgroup$ – lulu Jun 20 '16 at 20:40
  • $\begingroup$ Assuming spherical turkeys of identical mass... $\endgroup$ – Joffan Jun 20 '16 at 20:42
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Let the missing digits be $a$ and $b$ in that order; we know that $a679b$ is divisble by $8$. Since $a6000=a6\cdot1000$ is divisible by $8$, this means that $79b$ is divisible by $8$. $720$ is a multiple of $8$, so $79b-720=7b$ is a multiple of $8$, and the only possible choice for $b$ is clearly $2$.

Now we know that $a6792$ is divisible by $9$. This means that the sum of its digits is a multiple of $9$; that sum is $a+6+2\cdot9$, so $a$ must be $3$. The original amount must therefore be $\$367.92$.

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Using your realisation we get a679b (where a,b are digits between 0 and 9) is divisible by 72 and hence by 8 and by 9.
Working modulo 8:
$790 + b \equiv 0$ mod 8
$-2 + b \equiv 0$ mod 8 and thus $b=2$.
Now the division rue for 9 gives $a+b+22$ is divisible by 9 $\implies$ a+6 is divisible by 9, so $a=3$.
Collecting this information, the total bill was \$367.92 and each turkey cost \$5.11.

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