Yes, you are right it can be any number but in this case it's 1. Let's see :
Consider longer wavelength making n fringes then shorter wavelength will make n+x fringes( where X is any integer equal to or greater than 1). Now we see that in our case how it comes out to be 1.
600(n)=450(n+x)
600n-450n=450x
15n=45x
n=3x
Since n has to be the least value and an integer(>0) and is proprtional to x ,we can guess the value of x, It must be such that it should give least value for n and must be an integer therefore in this case it comes out to be 1. One may say why x must be integer . This is so because fringes are integers.(or else other wavelength fringe will be n+x and become a non integer which wrong) If wavelengths were prime then also we can aplly same logic to get the value of x. In those cases fringes will be equal to others wavelength, as pointed out by John Rennie.
Consider the example where wavelengths are 11 and 17,
17(n)=11(n+x)
6n=11x
n=11x/6
To make n least integer such that x is also an integer we get x=6
So n_11=(n+x) = 11+6= 17 and n_17=11
Now you can see why we don't take x any fraction.