The first thing you might want to ask is how many factors there are.
According to counting principles, you can look at the exponents of the primes to decide. Since five, x, and y are all primes, and each has a power of 1, we can simply use the rule (p+1)(q+1)(r+1), where p, q, are the exponents of the primes. In this case, the 5, the x and the y are all primes to begin with. Also, all of the exponents are 1:
5¹x¹y¹
(1+1)(1+1)(1+1) = 8
This is a simple mental technique to get the number of factors.*
Obviously, 1 is a factor, and so is the original expression. Those are your first two.
Look what's, 5 is a factor, x is a factor, and y is a factor. Those are three more.
We can also pair the 5 with the x: 5x, the 5 with the y: 5y, and the x with the y: xy. Those are the last three.
So your 8 factors are:
1, 5, x, y, 5x, 5y, xy, and 5xy.
*Note that, if you are given 50x³y², things would be much more complicated because the prime factorization is 5² 2¹ x³ y² so that the exponents are 2,3,1,2
This means that there are a lot more factors:
(2+1)(3+1)(1+1)(2+1) = 72 factors!!!
Making the list would take a long time, but at least you know how many you'll have in the end.
