Suppose that cats are only sold in batches of 3 or k. Is it ways true that there's some n0 such that if n ≥ n0, you can purchase exactly n cats? When such an n0 exists, how does it depend on k? What if cats were sold in batches of a or b?

It isnt always true. It is only true when k is not a multiple of 3. If cats were sold in batches of a and b, then they have to be coprime, in other words, its only positive common divisor should be 1.

Step-by-step explanation:

If k is a multiple of 3, then any combination of batches you bought will give as a result a multiple of 3. Thus, you cant but, lets say 31 cats, or 301, or 3001, and so on.

If k is not a multiple of 3. Then k and 3 are coprime, which means that there exists n and m such that 3n + mk = 1.

Thus,

3n + mk = 1

6n + 2mk = 2

One either m or n is negative. If, for example, n is negative, then, we will be able to form any number from - 3*2n = - 6n (which is positive) onwards, because

-6n + 1 = - 3*2n + (3n+mk) = 3 * (-n) + mk

-6n + 2 = - 6n + (6n + mk) = mk

And any other number greater than - 6n+2 is obtained either from 6n, 6n+1 or 6n+2 by adding a positive multiple of 3.

For m negative the argument is similar.

If cats were solver in batches of a or b, then we can only get cats that are a multiple of the greater common divisor of a and b. If that greater common divisor is 1 (in other words, a and b are coprime), then, we can obtain any number large enough.