It’s common for the Putnam exam to include problems that involve the number of the current year in some way. (There weren’t any such problems on the past exam, but in 2011 there were two, A1 and B4.) Usually these problems don’t rely on any specific property of the number, but they simply use the number as a stand-in for “n”, so as to make the problem a little bit more concrete; for example, for either of the 2011 problems, you could replace the number 2011 with any other number without changing the nature of the problem.
But, occasionally, there are problems that do rely on specific properties of the year number. It may be something simple, like whether it’s odd or even. If the year number is prime (which has been true more often than you might expect in recent years), it could use that. Let me be the first to predict that the 2025 exam will take advantage of the fact that 2025 is a square.
So it’s helpful to know the prime factorization of the year: 2013 = 3*11*61. It doesn’t take up that much space in your brain, and, even though it’s not so likely, it could possibly turn out to be useful for this year’s exam. In any case, it’s a good party trick to pretend you just did it in your head.