Since 3.62 can be divided evenly by .1 does that mean it is a muliple of .1?

Question

Medicare was billed 3.62 mg of a drug expressed with a code description of .1 mg per unit (in other words we billed 37 units of the code, because we rounded up). Medicare says if the dosage given is not a multiple of the number provided in the code description, the provider shall round up to the nearest whole number in order to express then number as a multiple. I'm not sure if we should round up or down. I think 3.62 is a multiple of .1 but when you divide 3.62 by .1 there's a remainder of .2. Even though there's a remainder it is evenly divisible. So I'm confused.

Answer 1

That translates to a multiple

$$ k = \lceil m / 0.1 \text{mg} \rceil $$ and a dosage $$ m' = k \cdot 0.1 \text{mg} $$

For your example $m = 3.62 \text{mg}$ we get $k = 37$ and $m' = 3.7 \text{mg}$.

The remainder $0.02 \text{mg} = (1/5) \cdot 0.1 \text{mg}$ was rounded up to a full unit of $0.1 \text{mg}$.