Next we're going to look at divisibility rules, which are really just little tricks that help us figure out if one number is divisible by another. Let's take a look at this question. Is six a factor of 7152? This is another way of saying is 7152 divisible by six? Well, there's different ways we can figure this out. We could divide using long division and see if we end up with a remainder or not. We could also divide using a calculator. Or we can use divisibility rules. So divisibility rules are great if we want to do something that's a little faster than long division. Maybe we don't have or aren't allowed to use a calculator, we can always use these rules to help us out. So divisibility rules help us to see if a number is a factor of another number without having to use division. Now remember, a number is a factor if it divides into the other number with no remainder left over. Ten is a factor of another number if the number ends in zero. So here are some examples: 30, 250, 900. As long as there is a zero at the end, then ten is a factor. 100 is a factor if the number ends with two zeros. Here are some more examples. 500 901,200 as long as it ends in two zeros, then 100 is a factor. So we can tell if ten or 100 is a factor of another number by just looking at it. Two is a factor if the number ends in 0, 2, 4, 6 or 8. In other words, this applies to all even numbers. So if the number is even, then two is a factor. And here are some examples. 38, 194, 50 they have many different examples. But if it's even, then two is a factor. Five is a factor. If the number ends in zero or five, for example 45, 230, 7,005. As long as the last digit is a zero or a five, then five is a factor. Next, we're going to see the divisibility rule for three. Now, so far, all the examples that we saw only required us to look at the end of the number at the last digit to see if a particular number was a factor or not. To see if three is a factor, we have to do something a little differently. We have to take the sum of the digits and see if that is divisible by three. So that involves adding up all the digits of the number. And if that total is divisible by three, then the original number is divisible by three. So here are some examples. Let's look at the number 225. If we add up all the digits, two plus two plus five, it gives us nine. And nine is divisible by three. So that means our original number, 225 is divisible by three. Here's another one 1002. If we add up all those digits, we get three. So 1002 is divisible by three, and the number 87. The sum of its digits is 15, and 15 is divisible by three. So 87 is also divisible by three. To see if nine is a factor of a number, we do a process that's very similar as what we just did with the number three. So we want to find the sum of the digits and see if that is divisible by nine. For example, 576, we find the sum of its digits, five plus seven plus six, we get 18, and 18 is divisible by nine. So that means that our original number, 576, is divisible by nine. Let's look at our next example. 3123. We add up all of its digits and we get nine. That means that 3123 is divisible by nine. And even this number, 42,867, we add up each digit in that number and its total is 27, which is divisible by nine. So our original number is also divisible by nine. To see if six is a factor of a number, we need to see if both the numbers two and three are factors. So if two and three are factors of the number, then six is also a factor of a number. So that means that the number has to be even. Remember that all even numbers are divisible by two. And the trick for the three being a factor has to work as well. So we have to add up all the digits and make sure that that sum is divisible by three. So if both of those rules work, then six is a factor of that number. Now let's go back to our original question. Is six a factor of 7152? So remember, if six is a factor, that means that both two and three are factors. So let's see if this number passes both of those tests. First, to see if two is a factor, we need to see if the number is even. So we look at the last digit, and as long as the last digit is 0, 2, 4, 6, or 8, then it's even. Our last digit is two, so it passes that test. Now let's see if it passes the test for the number three. Let's see if the sum of the digits is divisible by three. We add up those digits, seven plus one plus five plus two, and we get 15, which is divisible by three. So that means that our number is also divisible by three. So it passes both tests. It's divisible by two and three. So that means that it's also divisible by six. So yes, six is a factor of this number. So there you go. You learned all different kinds of divisibility rules to see if different numbers are a factor of another number without even having to use division.