How to solve fizzbuzz

I was on a phone screen today and was given the famous (or infamous) FizzBuzz problem. I will provide a strategy to solve the class of problems that fizzbuzz represents, namely, take two numbers, a and b both positive. The question is phrased thusly:

Generate numbers 1 to n, print "fizz" for every number divisible by a, "buzz" for every number divisible by b, and "fizzbuzz" for every number divisible by a*b. If not, print the number itself.

The key here is to start with the most restrictive case first, the a*b, before tackling the larger of a and b as an else if clause and the default case being the final line of the loop. Hence, in pseudocode:

for number in range(1, infinity):
if number % (a*b) == 0:
console_print 'fizzbuzz'
else if number % b:
console_print 'buzz'
else if number % a:
conosle_print 'fizz'
else:
console_print number


This code isn't in any language, but is probably easy-enough to translate to whatever language you need to. Also, it is general enough to (hopefully) enumerate the general case. I do, however, welcome further questions and feedback. Many thanks!