# How Much for a Turkey?

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The following problem is a bit of a classic. It can be found in George Polya's How to Solve It:

Among Grandfather's papers a bill was found:

72 turkeys \$_67.9_

The first and last digit of the number that obviously represented the total price of those fowls are illegible and are replaced here by blanks. What are the two faded digits and what was the price of one turkey?

## Solution

Assume that each of the 72 turkeys cost the exact same (i.e. the turkeys weren't sold by weight) and that price is an integer number of cents. Therefore, _679_ must be divisible by 72. If a number is divisible by 72 it must be divisible by both 8 and 9. Using the tests for divisibility, a number is divisible by 8 only if one of the two hold:

1. The hundreds digit is even and the last two digits are a multiple of 8.
2. The hundreds digit is odd and the last two digits are a multiple of 4 but not 8.

Since the hundreds digit is odd, the last two digits must be a multiple of 4 but not 8. The only number to put in the second blank to make this true would be 2. We now have _6792.

Now, the number _6792 must be divisible by 9, so its digits must add to a multiple of 9. To accomplish this, the first blank must be 3. Therefore, the 72 turkeys cost \$367.92 and each turkey cost \$5.11.