Thread: Divisibility

I know I am supposed to use Divisibility Theorems to solve this problem, but I am not sure how to go about doing it.

An old receipt has faded. It reads 88 chickens at a total cost of x4.2y$ where x and y are unreadable. How much did each chicken cost?

Originally Posted by RonyPony

I know I am supposed to use Divisibility Theorems to solve this problem, but I am not sure how to go about doing it.

An old receipt has faded. It reads 88 chickens at a total cost of x4.2y$ where x and y are unreadable. How much did each chicken cost?

Assume a chicken costs an integer number of cents.

Then x42y is divisible by 8 and 11. y must be even, and we can use the test for divisibility by 11 using digits: x - 4 + 2 - y is divisible by 11. That is, x - y - 2 is divisible by 11. It can be seen that 0 is the only attainable number, so x-y-2=0 and we try

(x,y) = (2,0)
(x,y) = (4,2)
(x,y) = (6,4)
(x,y) = (8,6)

The only one that works is (x,y) = (6,4), so we have $64.24 and each chicken costs 73 cents.

Originally Posted by undefined

That is, x - y - 2 is divisible by 11. It can be seen that 0 is the only attainable number, so x-y-2=0 and we try

I'm not sure what you mean by that or how you got it, can you explain it in a little more detail.

Originally Posted by RonyPony

I'm not sure what you mean by that or how you got it, can you explain it in a little more detail.

Divisibility Tests

Hello, RonyPony!

A slight variation of undefined's solution.

An old receipt has faded.
It reads 88 chickens at a total cost of . , where and are unreadable.
How much did each chicken cost?

88 chickens cost cents.
Hence, is divisible by 8 and 11.

A number is divisible by 8 if its last three-digit number is divisible by 8.
. . That is: . must be divisible by 8.
Since is a digit, the only choice is: .
. . And we have: .

A number is divisible by 11 if: .
. .
That is: .
. . and we have: .
Since is a digit: .

Hence: .

Each chicken cost: .