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!
Best Answer
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 $150$.
Hope this has convinced you!