No, no, 1 mi = 1.609 km.
is what you want to evaluate.
You're confusing with the conversion 1 mile = 5,280 feet.
There is no need to divide by anything in this case. 5266.26 km is the answer.
If you have a conversion that does not involve 1, like 16 oz. = 453.6 g or 12 in. = 30.48 cm, then yes, would end up doing a division. For example, convert 2,000 g to oz.:
... you would multiply 2,000 by 16, and then divide by 453.6.