A good way to visualize multiplication is as a rectangle.
3*5
* * * * *
* * * * *
* * * * *
If you turn the rectangle sideways, you change which edge is which, but the number of things hasn't changed.
5*3
* * *
* * *
* * *
* * *
* * *
Which is why 5*3 = 3*5, and in general a*b = b*a.
If you do the process in reverse and take a number, form a rectangle out of it, then take the sides and form rectangles out of the sides, and then take those rectangles and repeat as long as you can, you eventually run into numbers that can't be formed into rectangles. Those numbers are called primes, and since you can take a number that can be formed into a rectangle and work backwards like this, every number is either a prime or a bunch of primes multiplied together.
3 is prime:
*
* *
So is 5:
* *
* * *
You can make a rectangle out of 30 like this:
* * * * *
* * * * *
* * * * *
* * * * *
* * * * *
* * * * *
Which has sides of length 5 and 6.
5 is prime, but 6 isn't:
* * *
* * *
6 is 2 * 3, and both 2 and 3 are prime, so we're done, and 30 = 2*3*5, which is called the prime factorization of 30.
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"A dead thing can go with the stream, but only a living thing can go against it." --G. K. Chesterton