Finding factors and multiples

Which of the following numbers is a factor of 154? So when a number is going to be a factor of 154 is if we can divide that number into 154 and not have a remainder. Or another way of thinking about it-- a number is a factor of 154 if 154 is a multiple of that number. So let's look at each of these and see which of these we can rule out or say is a factor. So does 3 divide evenly into 154? Or, another way of thinking about it, is 154 a multiple of 3? Well you'll later learn that you could actually test whether something is divisible by 3 by adding up the digits. And if that's divisible by 3, then it's going to be divisible by 3. And so you see here, 1 plus 5 is 6. 6 plus 4 is 10. 10 is not divisible by 3. But if you didn't want to do that little trick-- and we have other videos where we go into more detail about that trick-- you can actually just divide 3 into 154. 3 doesn't go into 1. It does go into 15 five times. 5 times 3 is 15. Subtract. Then we have 0. Then you bring down a 4. 3 goes into 4 one time. 1 times 3 is 3. Subtract, and you have a remainder of 1. So 3 is clearly not a factor of 154, so we can rule that out. Now what about 5? Well, any multiple of 5 is either going to have 5 or 0 in the ones place. You see that if we write 5 times 1 is 5, 5 times 2 is 10, 5 times 3 is 15, 5 times 4 is 20, you either have a 5 or a 0 in the ones place. This does not have 5 or a 0 the ones place, so it's not going to be divisible by 5. 5 is not a factor. 154 is not a multiple of 5. Now 6 is interesting. You could do the same thing. You could try to divide 6 into 154. But if something is divisible by 6, it's definitely going to be divisible by 3 as well because 6 is divisible by 3. So we can immediately rule this one out as well. Because 154 is not divisible by 3, it's also not going to be divisible by 6. And you could try it out if you like. And we could make the same argument for 9. If something is divisible by 9, it's going to be divisible by 3 because 9 is divisible by 3. Well, it's not divisible by 3, so we're going to rule out 9 as well. So we've ruled out everything. It looks like 14 is our only option, but let's actually verify it. Let's actually divide 14 into 154. 14 doesn't go into 1. It goes into 15 exactly one time. 1 times 14 is 14. We subtract. We get 1. Bring down the 4. 14 goes into 14 one time. 1 times 14 is 14. And of course, we have no remainder. So 14 goes into 154 exactly 11 times. Or 11 times 14 is 154. 154 is a multiple of 14. Let's do one more. Which of the following numbers is a multiple of 14? So now we have 14, and we're trying to think of its multiples. So there's two ways of doing this. You could go number by number and try to divide 14 into them, or we could just try to figure out what all of the multiples of 14 actually are. So let's try to do that. Let's try to do that second technique. 14 times 1 is 14. You add another 14. 14 times 2 is 28. Add another 14. Let's see, you add 10. You get to 38. Then you add 4 more. You get to 42. Then you add another 14. I haven't seen any of these numbers show up yet. Add another 14 to this. You get to 56-- still not quite there. Add another 14. Let's see, if you add 4, you get to 60, and you have to add the 10. So then you get to 70. And it looks like we have found one of these numbers. 70 is a multiple of 14. 14 times 1, 2, 3, 4, 5-- 14 times 5 is 70.