# Clarification on meaning of question (ISEE Test Upper Level)

QUESTION:

$$a$$ is a factor of $$10$$ and $$b$$ is a factor of $$15$$.

Column A Column B
The smallest value that ab must be a factor of 30

Is Column A greater than/ smaller than / equal to Column B?

The possible values of $$a$$ are $$1,2,5$$ (and maybe $$10$$ itself?). The possible values of $$b$$ are $$1,3,5$$ (and maybe $$15$$?). So we could have $$ab$$ equal to $$1,2,3,5,6,10,15,25$$ and maybe a bunch more if $$10$$ and $$15$$ are included. So the smallest value that $$ab$$ could be a factor of, well, wouldn't that be $$1$$? or maybe $$6$$ if $$1$$ is not allowed? Either way, Column A would be smaller than Column B. But the solution states the opposite. I don't think I understand the question. Help appreciated!

The answer is $$150$$ which is greater than $$30$$, we must include the case when $$a=10$$ and $$b=15$$ (since the question doesn't explicitly ask us to use only non-trivial divisors). A justification of the above statement is as follows:
The set of all possible values for the product $$ab$$ is $$\{1,2,3,5,6,10,15,25,30,50,75,150\}$$. Now suppose the smallest value that $$ab$$ must be a factor of were less than $$150$$, take for instance that it was $$75$$, then if $$ab=150$$, we'd have the contradiction that $$150$$ is not a factor of $$75$$. We can extend this line of reasoning to any other number $$<150$$. Thus the only number that satisfies the requirements is $$150$$.