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Finding factors of a number

Sal finds the factors of 120. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

Find all of the factors of 120. Or another way to think about it, find all of the whole numbers that 120 is divisible by. So the first one, that's maybe obvious. All whole numbers are divisible by 1. So we could write 120 is equal to is to 1 times 120. So let's write a factors list over here. So this is going to be our factors list over here. So we just found two factors. We said, well, is it divisible by 1? Well, every whole number is divisible by 1. This is a whole number, so 1 is a factor at the low end. 1 is a factor. That's its actual smallest factor, and its largest factor is 120. You can't have something larger than 120 dividing evenly into 120. 121 will not go into 120. So the largest factor on our factors list is going to be 120. Now let's think about others. Let's think about whether is 2 divisible into 120? So there's 120 equals 2 times something? Well, when you look here, maybe you immediately recognize that 120 is an even number. It's ones place is a 0. As as long as its ones place is a 0, 2, 4, 6 or 8, as long as it's an even number, the whole number is even and the whole number is divisible by 2. And to figure out what you have to multiply by 2 to get 120, well, you can think of 120 as 12 times 10, or another way to think about it, it's 2 times 6 times 10, or 2 times 60. You could divide it out if you want. You could say, OK, 2 goes into 120. 2 goes into 1 no times. 2 goes into 12 six times. 6 times 2 is 12. Subtract. You get 0. Bring down the 0. 2 goes into 0 zero times. 0 times 2 is 0, and you get no remainder there, so it goes sixty times. So we have two more factors right here. So we have the factors. So we've established the next lowest one is 2, and the next highest factor, if we're starting from the large end, is going to be 60. Now let's think about three. Is 120 equal to 3 times something? Well, we could just try to test and divide it from the get go, but hopefully, you already know the divisibility rule. To figure out if something is divisible by 3, you add up its digits, and if the sum is divisible by 3, we're in business. So if you take 120-- let me do it over here. 1 plus 2 plus 0, well, that's equal to 1 plus 2 is 3 plus 0 is 3, and 3 is definitely divisible by 3. So 120 is going to be divisible by 3. To figure what that number that you have to multiply by 3 is, you could do it in your head. You could say, well, 3 goes into 12 four times, and then you-- well, let me just do it out, just in case, just for those of you who want to see it worked out. 3 goes into 12 four times. 4 times 3 is 12. You subtract. You're left with nothing here. You bring down this 0. 3 goes into 0 zero times. 0 times 3 is 0. Nothing left over. So it goes into it forty times. And the way to think of it in your head is this is the same thing as 12 times 10. 12 divided by 3 is 4, but this is going to be 4 times 10, because you have that 10 left over. Whatever works for you. Or you can just ignore the 0, divide by 3, you get a 4, and then put the 0 back there. Whatever works. So we have two more factors. At the low end, we have 3, and at the high end, we have a 40. Now, let's see if 4 divisible into 120. Now we saw the divisibility rule for 4 is you ignore everything beyond the tens places and you just look at the last two digits. So if we're going to to think about whether 4 is divisible, you just look at the last two digits. The last two digits are 20. 20 is definitely divisible by 4, so 120 will be divisible by 4. 4 is going to be a factor. And to figure out what we have to multiply 4 by to get 120, you could do it in your head. You could say 12 divided by 4 is 3, so 120 divided by 4 is 30. So we have two more factors: 4 and 30. And you could work this out in long division if you want to make sure that this works out, so let's keep going. And then we have 120 is equal to-- is 5 a factor? Is 5 times something equal to 120? Well, you can't do that simple-- well, first of all, we could just test is it divisible? And 120 ends with a 0. If you end with a 0 or a 5, you are divisible by 5. So 5 definitely goes into it. Let's figure out how many times. So 5 goes into 120. It doesn't go into 1. It goes into 12 two times. 2 times 5 is 10. Subtract. You get 2. Bring down the 0. 5 goes into 20 four times. 4 times 5 is 20, and then you subtract, and you have no left over, as we expect, because it should go in evenly. This number ends with a 0 or a 5. Let me delete all of this so we can have our scratch space to work with later on. So 5 times 24 is also equal to 120, we have two more factors: 5 and 24. Let me clear up some space here because I think we're going to be dealing with a lot of factors. So let me move this right here. Let me cut it and then let me paste it and move this over here so we have more space for our factors. So we have 5 and 24. Let's move on to 6. So 120 is equal to 6 times what? Now, to be divisible by 6, you have to be divisible by 2 and 3. Now, we know that we're already divisible by 2 and 3, so we're definitely going to be divisible by 6, and you should hopefully be able to do this one in your head. 5 was a little bit harder to do in your head. but 120, you could say, well, 12 divided by 6 is 2, and then you have that 0 there, so 120 divided by 6 would be 20. And you could work it out in long division if you like. So 6 times 20 are two more factors. Now let's think about 7. Let's think about 7 here. 7 is a very bizarre number, and just to test it, you could think of other ways to do it. Let's just try to divide 7 into 120. 7 doesn't go into 1. It goes into 12 one time. 1 times 7 is 7. You subtract. 12 minus 7 is 5. Bring down the 0. 7 times 7 is 49, so it goes into it seven times. 7 times 7 is 49. Subtract. You have a remainder, so it does not divide evenly. So 7 does not work. Now let's think about 8. Let's think about whether 8 works. Let's think about 8. I'll do the same process. Let's take 8 into 120. Let's just work it out. And just as a little bit of a hint-- well, I'll just work it out. 8 goes into 12-- it doesn't go into 1, so it goes into 12 one time. 1 times 8 is 8. Subtract there. 12 minus 8 is 4. Bring down the 0. 8 goes into 40 five times. 5 times 8 is 40, and you're left with no remainder, so it goes evenly. So 120-- let me get rid of that. 120 is equal to 8 times 15, so let's add that to our factor list. We now have an 8 and now we have a 15. Now, is it divisible by 9? Is 120 divisible by 9? To test that out, you just add up the digits. 1 plus 2 plus 0 is equal to 3. Well, that'll satisfy our 3 divisibility rule, but 3 is not divisible by 9, so our number will not be divisible by 9. So 9 will not work out. 9 does not work out. So let's move on to 10. Well, this is pretty straightforward. It ends in 0, so we will be divisible by 10. So let me write that down. 120 is equal to 10 times-- and this is pretty straightforward-- 10 times 12. This is exactly what 120 is. It's 10 times 12, so let's write those factors down. 10 and 12. And then we have one number left. We have 11. We don't have to go above 11, because we already went through 12, and we know that there aren't any factors above that, because we were going in descending order, so we've really filled in all the gaps. You could try 11. We could try it by hand, if you like. 11 goes into 120-- now you know, if with you know your multiplication tables through 11, that this won't work, but I'll just show you. 11 goes into 12 one time. 1 times 11 is 11. Subtract. 1, bring down the 0. 11 goes into 10 zero times. 0 times 11 is 0. you're left with a remainder of 10. So 11 goes into 20 ten times with a remainder of 10. It definitely does not go in evenly. So we have all of our factors here: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120. And we're done!