Multiply-with-carry

Imagine you have a box of blocks with numbers on them, and you want to create a list of numbers that never repeats. To do this, you use a special machine called "multiply-with-carry".

First, you pick a starting number, which is like picking a block from the box. Let's say you start with the number 5.

Next, you pick two special numbers called "multiplier" and "carry". The multiplier is like a magic number that helps create unique new numbers, and the carry is like a reminder that helps keep track of the numbers you've already used. For this example, let's use a multiplier of 7 and a carry of 2.

Now, you put your starting number (5) into the machine, and you multiply it by the multiplier (7).

5 x 7 = 35

The answer is 35, but we're not done yet. We also add in the carry (2), which gives us:

35 + 2 = 37

So 37 is our first new number. We write it down on our list of numbers that never repeat.

Now, we take that number (37) and put it back into the machine, and multiply it by the multiplier (7).

37 x 7 = 259

Then, we add in the carry (2), which gives us:

259 + 2 = 261

So 261 is our second new number. We add it to our list.

We keep doing this over and over again, using the previous result as our starting number. Each time we generate a new number, we add it to our list. And we keep using the same multiplier and carry.

The cool thing about multiply-with-carry is that it creates a list of numbers that are always unique and never repeat. It's kind of like magic! But really, it's just a math trick that helps us create lists of numbers.