# Thread: primes and divisibility

1. ## primes and divisibility

I am having trouble in my new math course understanding what is going on. i do not know where to start on this problem i attached

2. Hello, ihavvaquestion!

This more of a math puzzle then a "real" math problem.

Someone spilled ink on a bill for 36 sweatshirts.
The result looked like the following.
There is one digit missing in front of the 8 and one digit missing after the 9.
The cost per shirt was not a frational part of a penny.

How much did each sweatshirt cost?
Find all possible costs per sweatshirt.

. . $\displaystyle \begin{array}{cccccccc} & ? & ? & . & ? & ? && \text{unit price}\\ & \times & & & 3 & 6 && \text{no. shirts}\\ \hline ? & 8 & 3 & . & 9 & ? && \text{Total}\end{array}$

Since the Total is a multiple of 4, it must end in a two-digit number divisible by 4.
. . There are two choices: .$\displaystyle 92\,\text{ or }\,96$

Since the Total is a multiple of 9, the sum of its digits must be a multiple of 9.

If the Total is $\displaystyle ?83.{\color{red}92}$, the Total must be $\displaystyle {\color{red}5}83.92$

. . Then: .$\displaystyle \frac{583.92}{36} \:=\:16.22$

If the Total is $\displaystyle ?83.{\color{red}96}$, the Total must be $\displaystyle {\color{red}1}83.96$

. . Then: .$\displaystyle \frac{183.96}{36} \:=\:5.11$

Therefore, the unit price is either: .$\displaystyle {\color{blue}\$16.22}\,\text{ or }\,{\color{blue}\$5.11}$

3. 2 questions...

how do you know the total is a multiple of four? is it because 36 is divisible by 4?

also, what total is divisible by 9? the 36 shirts?