Greatest Common Divisor parking meter problem

Question

A parking meter can hold $k$ quarters, $2k$ nickels, and $4k$ dimes. Find all $k$ such that the total when the meter is full is a whole number of dollars.

Can anyone point me in the right direction? I don't know where to start

Answer 1

We can start off by writing the total value as an expression in $k$:

$$t=0.25k+2(0.05)k+4(0.10)k$$

Simplifying gives:

$$t=0.75k=\frac{3k}{4}$$

Hence, for $t$ to be an integer, $4\mid3k$. As $\gcd(3, 4)=1$, this is equivalent to $4\mid k$.

You can check that all these solutions work as we can let $k=4l$:

$$t=0.75(4l)=3l\in\mathbb{N_0}$$

Answer 2

How much money is in the meter if $k=1$? What is the smallest number you can multiply that by to get a whole number of dollars?