Q: Please give us the rest of the factorizing steps!
A: i can't give them all at once, because there's way too many. but let's pick a random number up to a million and i'll try to tell you the first 3 things i would do. The number we're working with today is 306124. The first thing is, notice it's even. How do we know this? because the last digit is even. so first we gotta cut the whole thing in half. this can be done by slicing it where you see even numbers that you can handle. so for me that's 30 6 12 4. cut those in half, and keep them in the same order, preserving the digits. so that makes 15 3 06 2.... or 153062. Next, notice it's still even. do the same thing again. this gets tricky because you have a lot of consecutive odd numbers in there. cutting numbers in half when they have a lot of odd digits, without paper, can be a little annoying. but if you can handle the 1530, then the 62 part is easy. 1500 in half is 750, plus the 15 is 765. and the 62. so 76531. Now it's odd and you can't divide it anymore. So you can go ahead and put some 2's in your final answer, because you divided it by 2, twice. so you know that your answer is going to include 2 twice. Now, you're left with an odd number: 76531. The next step is to check if it's divisible by 3. The trick for this is, and most people seem to know this trick, if the number is divisible by 3, all the digits are gonna add up to a multiple of 3. so you can group them. so, looking at 76531, .. knock out the 6, i can knock out 7 and 5 (75, mult of 3, or 57, either way, same thing) knock out the 3, ... and you're left with 1 which is non-3. so by knocking out digits and groups of digits which add to 3 or mult of 3, if all the digits got knocked out that way then we know it's a mult of 3, but if we were left with any non-3s, (which we were), then we know it's not a mult of 3. Ok, now the real game begins. We're going to check every other prime number, in ascending order. No need to check 5, because we can see that it's not a multiple of 5 (hopefully you can see that outright ... but the way to tell that is if the last digit is 0 or 5, which it's not). So next we're going to check 7. Every prime we're going to check from here on out will be based on the same system. We're going to manipulate the main number (76531) by adding or subtracting the checker as many times as we want. IF the main number is a mult of 7, then that means we can add or subtract 7 as many times as we want, and it should STILL be a mult of 7. We can ALSO DIVIDE the main number by ANYTHING and it should still be a mult of 7, if it originally was. So basically the idea is to do these manipulations in a way that gets you to something you recognize as a mult of 7 or not. There are many ways of doing this. First, if you want, you can just drop the 7 from the left. because this essentially would mean you're subtracting 70000. which, if the original number was a mult of 7, then it should still be a mult of 7 when you subtract 70000. so we'll knock out the 7, and you have 6531. Next, i'm going to do probably the most common technique, which is to work from the right. Here's an interesting principle: if we've already reduced the main number in the ways we've already reduced it (cut it in half until it's odd, divide by 3 until it's not a mult of 3, divide by 5 until it's not a mult of 5), then THAT MEANS that for ANY future prime number you wanna check, you can ALWAYS cause the number to level off with a zero at the end by either adding or subtracting THE CHECKER OR ITS TRIPLE. That's because all your future prime numbers that you're gonna check are gonna end in either 1,3,7, or 9. So that means those are always either 1 or 3 away from 10. And if you've already reduced your main number properly, then that number will ALSO always be either 1 or 3 away from the nearest 10. So you just pick which maneuver to make based on what you need. So looking at our main number as it stands now, 6531, and if we're checking for 7, i can tell you that what you wanna do is subtract 21. That's how you can get it to level off to a zero at the end. 6510. and of course you can drop the zero, because that does NOT change whether or not the main number is divisible by 7. so that leaves you with 651. the same move again works: subtract 21, gets you to 630 and drop the zero, you have 63. Do you recognize it yet, as something that you know to be a multiple of 7, or something you know to not be a multiple of 7? you might recognize 63 as a multiple of 7. But in case you don't see that, you could make another move, by adding 7, and you get 70, drop the zero, is 7. Now do you recognize it (7) as either being a multiple of 7 or non-multiple of 7? i think you do. So we can confirm that 7 is one of the factors. next comes the not so fun part where you have to actually divide 76531 by 7. then when you get the result, continue working with this new main number, and you also have to check 7 again on that number in case it's in there twice