You can easily check if a factorization is correct using a conventional computer. Of course factorizing 15 is pretty useless in itself, but you have to start somewhere. To put things into perspective, assume you have a number with 1000 digits, and you want to factorize that. The best known conventional algorithm for doing that is the General Number Field Sieve with which the factorization would take in the order of 1.4 * 10^43 operations. Assuming you had a computer capable of executing a trillion operations per second it would still take about 4.6 * 10^23 years, which is 33 trillion times the age of the universe!

Now assume you had a quantum computer with enough qubits - we would need at least 3322 qubits. Let us say that it is otherwise a pretty crappy quantum computer as it only gets the factorization right 0.1% of the time. Now we try to use our quantum computer. It gives us an answer in the order of just a few billion operations. Even if it is quite slow and only capable of 1 million operations per second, it would still give an answer in less than an hour. This answer is probably wrong, however we can easily check that using our conventional computer. Checking if a number divides another is FAST. It can actually be done in slightly more than just the size of the input - the existence of a factor in a 1000 digit number would take the order of maybe 100,000 operations to check - in much less than a second.

So the time it takes to validate the answer is negligible here. We just keep on asking the quantum computer to try again until we get it right. So how long would it take? After 10000 tries we would have gotten the correct result with a probability of 99.995%. So if every try takes 1 hour, we would be pretty sure to have succeeded in less than a year (10000 hours = 1 year 1 month 21 days 6 hours). So even with this big but crappy quantum computer we would be able to factorize the integer in less than a year instead of 33 trillion times the age of the universe.