
Simple Number Theory
Positive integers 30, 72, and N have the property that the product of any two of them is divisible by the third. What is the smallest possible value of N?
Note I have not yet taken a Number Theory course.
I think I have found the solution using a bit of reasoning and some luck. N=60? I figured N could not be smaller than the gcd of 30 and 72, and could not be greater than their product. I also found a pattern for (30N)/72. Inputting 10, 15, 20, 30 for N gave a result of 25/6, 25/4, 25/3, 25/2, respectively. I figured that this converged to 25/1, which would then be my solution. N=60 indeed yields 25/1.
However, I feel that this is closer to luck than anything else, and also it is not very elegant. Can someone show me another way of doing this, perhaps something more elegant?
F

Your reasoning seems okay.
After noting that , I would probably list all of my possibilities at this point: 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 27, 30, 36, 40, 45, 48, 54, 60, 72, 80, 90, 108, 120, 135, 144, 180, 216, 240, 270, 360, 432, 540, 720, 1080, 2160.
Now, 30 divides . Since and , we must conclude that is a multiple of 5. This leaves 10, 15, 20, 30, 40, 45, 60, 80, 90, 120, 135, 180, 240, 270, 360, 540, 720, 1080, 2160.
On the other hand, 72 divides . By similar reasoning as before, must be a multiple of . Eliminating the bad possibilities leaves 60 as the smallest number satisfying all three conditions.

872 so for 8 to divide 30N, 4N. 530, thus 5N, since 5∤72. 3N since 972 and 9∤30. Thus N must be at minimum 2²·3·5 = 60.

Consider the three numbers
the set satisfies the requirement because
Let , so .
To minimize , we have to maximize so obviously what we are looking for is the greatest common divisor of and which is
Therefore


I really like this solution, it seems the more obvious to me. Thanks a lot everyone.