Math question I am stumped. A store manager priced a pair of earrings in dollars and cents such that...?

I'm puzzled by the wording of the question also.

For example, if the earrings were 96¢, then 4% would be $0.0384 but that would round up to 4¢. The result would be $1.00 as the total for the earrings and the sales tax. The only smaller number would be $0 which would imply the store manager was giving the earrings away and I'd hardly say they were being "sold".

Answer:
$1.00

UPDATE:

Alternatively, if the assumption is we aren't rounding the sales tax (odd assumption), then the final amount needs to be an exact multiple of 100 cents.

The greatest common factor of 104 and 100 [*correction 10000] is 8, so that would mean an original price of $100/8 = $12.50 resulting in a total price of $104/8 = $13.00

That's the first amount that works:
$12.50 * 1.04 = $13.00 (exactly, no rounding)

Alternate answer:
$13.00

UPDATE #2:

Let P be the integer price of the item in *cents*.
Let T be an integer representing the total cost in dollars.
100T will represent the total amount in cents.

To get from the item price P (in cents), we need to multiply by 1.04. This is mathematically equivalent to adding 4% of the item price.
--> 1.04P

The result should be the total amount in cents (100T)
1.04P = 100T

But to make this all work, we want everything in integers. We can do this by multiplying both sides by 100:
104P = 10000T

Reduce each by the greatest common factor of 104 and 10000. We can divide both sides by 8 because:
104 = 2^3 * 13
10000 = 2^5 * 5^5
GCD(104, 10000) = 2^3 = 8

13P = 1250T

The smallest positive integer solution to this occurs when P = 1250 and T = 13
13(1250) = 1250(13)

Item price = 1250¢ = $12.50
Total = $13

104 = 8*13
If n is the whole number of dollars after 4% sales tax was added, then,
100n/104 = 12.5n/13 ... and that is to be a reasonably small integer
Formula suggests n = 13 with pre tax price at $12.50

An engineer has entered the conversation

1 x 100/104 = lots of decimal places. Fail
2 x 100/104 = lots of decimal places. Fail
...
12 x 100/104 = lots of decimal places. Fail
13 x 100/104 = 12.5 Pass

Price $12.50
+ tax = £13

I think I got the answer.
If the sales price of the earrings is $12.50 then adding a 4% sales tax gives us $13 dollars exactly.
My reasoning was to solve (12 + x) * 1.04 = 13
There was some trial and error , I tried
(1 + x ) *1.04 = 2 , (2 + x) *1.04 = 3, etc.
In each case x was not a nice terminating decimal, until we hit 13.

Also notice that 13 is the smallest whole number such that
13 / 1.04 is a terminating decimal.

There might be an easier way to do this. though.

"when 4% sales tax was added, the result was a whole number of dollars"

$12.50 earrings + $0.50 tax = $13 total

$9.62 earrings plus .38 tax = $10.00

A store manager priced a pair of earrings in dollars and cents such that
when 4% sales tax was added, the result was a whole number of dollars and 0 cents.
Find the smallest possible number of dollars the items sold for, including the sales tax.
1.04x = y
x = y/1.04
For y = 13
x = 13/1.04 = 12.5
Price + Tax = $12.50 + $0.50