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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

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  • $\begingroup$ For those not based in the US these coins are 0.25 dollars, 0.05 dollars and 0.10 dollars respectively. $\endgroup$ – Ian Miller Nov 4 '15 at 1:35
  • $\begingroup$ Hint: What is the total as an expression in $k$? $\endgroup$ – Element118 Nov 4 '15 at 1:36
  • $\begingroup$ A quarter is of a different value from a nickel. The value of all the quarters is $0.25k$. Can you do the same for the other 2 denominations? $\endgroup$ – Element118 Nov 4 '15 at 1:41
  • $\begingroup$ @Element118 Ok so $t = .25k + 2(.05k) + 4(.10k)$? $\endgroup$ – hawk2015 Nov 4 '15 at 1:42
  • $\begingroup$ Hint: can you simplify this? What if it is a whole number of dollars? $\endgroup$ – Element118 Nov 4 '15 at 1:44
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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 $k=4l$:

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

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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?

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