*This problem follows on from What Numbers Can We Make?*

The interactivity below creates sets of bags similar to those in What Numbers Can We Make?

Investigate what numbers you can make when you choose three numbers from the bags. Once you think you know what is special about the numbers you can check your answer.

You can also find what is special when you choose four, five or six numbers.

When you have an efficient strategy, test it by choosing 99 or 100 numbers.

*Always enter the biggest possible multiple. The "plus" number may include zero.*

The answer is always
than a multiple of

Can you explain your strategies?

And finally...

If the bags contained 3s, 7s, 11s and 15s, can you describe a quick way to check whether it is possible to choose 30 numbers that will add up to 412?

There are a few related problems that you might like to try:

Shifting Times Tables

Charlie's Delightful Machine

A Little Light Thinking

Remainders

Where Can We Visit?

Cinema Problem