Question I find hard to comprehend and I don't understand the answer

[Noahnoah32]

In one class there are sixty notes numbered from 1 to 60. In another class, there are also sixty notes numbered from 1 to 60. Sixty people draw a random note from each box and multiply their two numbers together. If 6 goes up in the result, the person gets a soda. How many sodas are at least needed?

How do I comprehend this?

The Answer is 40

The reason

If a person draws two notes where 6 = 2 * 3 goes up in the result, then 3 must go up in at least one of the two numbers. There are only 20 notes in each box where 3 goes up. It is actually possible that 40 sodas will be needed: In each class, thre are 20 numbers where 3 goes up, combined with the 20 even numbers where 3 does not go up from the other box, then all these combinations give sodas. Similarly the other way around.

I don't understand the answer. 2 does also go up in 3 why can't I then say 60/2=30 so 2 goes up in 30 numbers times that with two like with three and that gives sixty and not 40. Can someone help me with this? Thanks for helping.

 

[Dr.Peterson]

If 6 goes up in the result

I think you mean, if 6 evenly divides the result. Am I right? (But then you are wrong that "2 does also go up in 3".)

I don't think the explanation is an adequate proof, because it doesn't mention multiples of 6 in either box.

The goal is to find the maximum possible number of products that are divisible by 6. In order to accomplish this, we would like to pair each of the 10 odd multiples of 3 in one box with an even number from the other, and pair each of the 10 even multiples of 3 in each box with anything left. This is the best we can do. It remains to show that this is possible.

Your attempt is less adequate, because you don't deal with multiples of 3 at all.