Alan has 17 7/8 gallons of gas. his mower's gas tank holds 3 1/4 gallons. how many times can he fill his mower

(17.875) / (3.14) = 5.693... (rounded)

IF Alan waits until his mower's tank is bone-dry empty before
adding any gas, then he will fill it 5 times, using

                         (5 x 3.14) = 15.7 gallons

of the precious golden fluid. Then, after it runs dry and dies the
fifth time, he will have only (17.875 - 15.7) = 2.175 gallons left in
the goat-skin, and although he won't be able to fill it to the rim,
he'll be able to replenish about 69% of it, which might get him
another front yard before the sun sets.

So I'll say that  5  is the LEAST number of times he will fill it.

But you asked "How many times CAN he fill ...".

There is no limit on the MOST number of times he can fill it.
If he wants to, he can top it off after every 5 minutes that it runs,
maybe adding 1 ounce every time to bring it up to the rim.

With 17-7/8 gallons, he could do that 2,288 times if he enjoys it !