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## Divisibility

Integer properties, this is one of the test's favorite questions to ask about, the integer properties. So this module we will be talking about all the properties of the integers, and of course the integers are the non-fraction numbers. They include all the positive numbers, positive whole numbers 1, 2, 3. They also include 0 and also negative whole numbers, -1, -2, -3, all that.

That's a set of integers and that's a good term to know, incidentally. You need, the test will expect you to know what an integer is. In fact, most of what we'll discuss in this module concerns only the positive integers. The ordinary counting numbers 1, 2, 3, 4, 5, 6, etc. So these are two specific categories of numbers we'll be focusing on this module.

Now of course, the test includes all kinds of numbers other than this, but we're gonna focus right now on these. In this particular video, we will discuss a few interrelated ideas, these are mathematical terms you definitely need to know and understand for the test. These are terms that will appear in test questions. So the big three to understand, first of all are factor, divisor, and divisible.

Let's start by discussing these terms with formulas and some simple numerical examples. In all these formulas, assume A, B, and C are positive integers. If we say that A*B = C, so the product of A and B equals C. Then we're saying that A and B are factors of C. In other words, we can multiply them together to get C.

To say that A is a factor of C is to say that we can multiply A by some other integer, and the product will be equal to C. So, for example, 3 is a factor of 6. 25 is a factor of 100. Notice that 1 is a factor of every positive integer. That's an important idea.

Notice also that every integer is a factor of itself. Thus, every positive integer greater than 1 has at least two factors, 1 and itself. So for example the number 84, whatever other factors it has, we know that one of the factors of 84 has to be 1 and one of the factors of 84 has to be 84. If C/A = B, so in other words, we can perform this division and the output of the division is still an integer, then we say that A is a divisor of C, because it divides evenly into C.

One number divides evenly into another when the quotient is an integer. We can also say that C is divisible by A. Notice there is absolutely no difference between factor and divisor. These two words mean exactly the same thing. So we use one more in the context of multiplication, one more in the context of division, but really, it means exactly the same thing.

Every factor is a divisor, every divisor is a factor. Thus, we have three interchangeable ways to say the same thing. And the test will use all three of these. We could say 8 is a factor of 24, 8 is a divisor of 24, and 24 is divisible by 8. All three of those say exactly the same thing.

They communicate exactly the same mathematical information. And so don't be confused, the test might use different wording in different questions. Similarly, 8 is not a factor of 12, 8 is not a divisor of 12, and 12 is not divisible by 8, because the quotient 12 divided by 8 is not an integer. In what amounts to the same thing, there's no positive integer B, such that the product 8*B would exactly equal 12.

This raises the interesting question, what happens when we divide 12 by 8? There are two separate but perfectly correct mathematical procedures. If two numbers divide evenly, then (integer) divided by (integer) equals (integer): the quotient is an integer, and that's it. That's very easy. But if the two items don't divide evenly, we have two possible options, each perfectly correct.

Option number one is to have an integer quotient and an integer remainder. So there we would say 12 goes into 8 once, a quotient of 1, with a remainder of 4. Option number two would be to express the quotient as a fraction or decimal. So we could say 12 divided by 8, that fraction simplifies to 3 over 2. We could also write that as a mixed numeral, one half, and we could write that as 1.5.

So both of these are perfectly correct. You will need to understand how to employ either option, but you will never have to decide which one to use. The nature of the test question will always make clear which one of them they mean. In other words, whether they're gonna be talking about remainders or whether they're gonna be talking about fractions and decimals as a result of division.

We discussed changing between fractions and decimals in the Arithmetic and Fraction module in the videos Conversions with Fractions and Decimals. So if that part is unclear to you, for example, if you haven't seen those videos, I would suggest go back and look at those videos. That's where we talk about how to turn a mixed number into an improper fraction, how to turn either into a decimal, that sort of thing.

Later in this module, we will discuss remainders at a greater level of sophistication. So, right now that was just a taste. We actually have a later video that are gonna be devoted exclusively to the idea of division with remainders. So that's coming up.

Now on another topic, suppose we had to find all the positive factors of 36. Now, a test question may ask how many factors does 36 have? There are also procedures, for example, when we get to factoring quadratics where it's gonna be important to find the factors of a number. Also, when we find the prime factorization, it will help to find some factors of the number.

But suppose we had to find all the factors. Well it's certainly clear, for example, we know 1 is a factor, 36 is a factor, maybe it's clear that 2,3, and 4 are factors, 7 is not a factor. We could kind of go haphazardly, but there's actually a system we can follow to find every factor. The easiest way is to list the factor pairs, the pairs of numbers that have a product of 36.

So the first pair I'm gonna list is 1 and 36. Those are clearly two factors, 1 is a factor of every number 36 has to be a factor of itself. Well 2 goes into 36, so that would be 2 times 18, that gives me two more factors. 3 goes into 36, that would give me 3 times 12. 4 goes into 36, that would be 4 times 9.

5 does not go into 36, so we'll skip that. And then we get to 6, well, 6 times itself, so we stop with six. Once you get to a number, you get to a number times itself, or you pass, and you reconnect with one of the numbers you have earlier in the list, something like that, then you know you're done. So now we know we are guaranteed that we have found every single factor of 36.

So, we don't count 6 twice. Notice that 36 has nine positive factors including 1 and itself. And so the procedure, again, you just list them in pairs. You skip the numbers. If any number, if the number's not divisible by that particular number, you just skip it.

And then you keep going until either you double back on yourself or you come to a number that's the product of itself. For small numbers, numbers less than 100, we can count the positive factors simply by making a list of factor pairs. We'll learn another, more efficient technique for larger numbers later in this module.

So for example, the test could give you a number like 12,600, and ask you to find, how many factors does this have? And it would just take far too much time to list all those factors. It would be, it would take like ten minutes to make a factor list. You wouldn't have time to do it on the test. Later on in this module we'll actually talk about an incredibly efficient procedure and we'll actually use that particular number, 12,600.

We'll find the number of factors in it very easily, so don't worry about that now. Just know that for numbers less than 100, you can just make the list of factor pairs. Finally, a note about negative integers. I've been focusing mostly on positive integers here.

Technically it is true that both +4 and -4 are factors of -12. Both +4 and -4 are divisors of -12. And -12 is divisible by both +4 and -4. It's true that every positive factor, for example 36 had nine positive factors, we could just put a negative sign in front of them. There'd be nine negative factors also.

So usually we're not as concerned with the negative factors, cuz they're just repeats of the positive factors. And that's why the test almost never asks about that or expects you to know it. But technically, factors and divisors could be negative. And that is something that we may have to watch out for on a more advanced question. In summary, we talked about the interrelated ideas of factors, divisors, and divisibility.

We talked about options for division that don't come out even, remainders versus non-integer quotients. Remember, remainders we'll be discussing in much greater detail a few modules from now. We talked about listing the factors of a number by listing factor pairs and introduced the idea of counting factors.

For larger numbers, we'll have another video later on. We'll talk about that in greater detail. And we talked about the seldom-used rules for negative factors.

Read full transcriptThat's a set of integers and that's a good term to know, incidentally. You need, the test will expect you to know what an integer is. In fact, most of what we'll discuss in this module concerns only the positive integers. The ordinary counting numbers 1, 2, 3, 4, 5, 6, etc. So these are two specific categories of numbers we'll be focusing on this module.

Now of course, the test includes all kinds of numbers other than this, but we're gonna focus right now on these. In this particular video, we will discuss a few interrelated ideas, these are mathematical terms you definitely need to know and understand for the test. These are terms that will appear in test questions. So the big three to understand, first of all are factor, divisor, and divisible.

Let's start by discussing these terms with formulas and some simple numerical examples. In all these formulas, assume A, B, and C are positive integers. If we say that A*B = C, so the product of A and B equals C. Then we're saying that A and B are factors of C. In other words, we can multiply them together to get C.

To say that A is a factor of C is to say that we can multiply A by some other integer, and the product will be equal to C. So, for example, 3 is a factor of 6. 25 is a factor of 100. Notice that 1 is a factor of every positive integer. That's an important idea.

Notice also that every integer is a factor of itself. Thus, every positive integer greater than 1 has at least two factors, 1 and itself. So for example the number 84, whatever other factors it has, we know that one of the factors of 84 has to be 1 and one of the factors of 84 has to be 84. If C/A = B, so in other words, we can perform this division and the output of the division is still an integer, then we say that A is a divisor of C, because it divides evenly into C.

One number divides evenly into another when the quotient is an integer. We can also say that C is divisible by A. Notice there is absolutely no difference between factor and divisor. These two words mean exactly the same thing. So we use one more in the context of multiplication, one more in the context of division, but really, it means exactly the same thing.

Every factor is a divisor, every divisor is a factor. Thus, we have three interchangeable ways to say the same thing. And the test will use all three of these. We could say 8 is a factor of 24, 8 is a divisor of 24, and 24 is divisible by 8. All three of those say exactly the same thing.

They communicate exactly the same mathematical information. And so don't be confused, the test might use different wording in different questions. Similarly, 8 is not a factor of 12, 8 is not a divisor of 12, and 12 is not divisible by 8, because the quotient 12 divided by 8 is not an integer. In what amounts to the same thing, there's no positive integer B, such that the product 8*B would exactly equal 12.

This raises the interesting question, what happens when we divide 12 by 8? There are two separate but perfectly correct mathematical procedures. If two numbers divide evenly, then (integer) divided by (integer) equals (integer): the quotient is an integer, and that's it. That's very easy. But if the two items don't divide evenly, we have two possible options, each perfectly correct.

Option number one is to have an integer quotient and an integer remainder. So there we would say 12 goes into 8 once, a quotient of 1, with a remainder of 4. Option number two would be to express the quotient as a fraction or decimal. So we could say 12 divided by 8, that fraction simplifies to 3 over 2. We could also write that as a mixed numeral, one half, and we could write that as 1.5.

So both of these are perfectly correct. You will need to understand how to employ either option, but you will never have to decide which one to use. The nature of the test question will always make clear which one of them they mean. In other words, whether they're gonna be talking about remainders or whether they're gonna be talking about fractions and decimals as a result of division.

We discussed changing between fractions and decimals in the Arithmetic and Fraction module in the videos Conversions with Fractions and Decimals. So if that part is unclear to you, for example, if you haven't seen those videos, I would suggest go back and look at those videos. That's where we talk about how to turn a mixed number into an improper fraction, how to turn either into a decimal, that sort of thing.

Later in this module, we will discuss remainders at a greater level of sophistication. So, right now that was just a taste. We actually have a later video that are gonna be devoted exclusively to the idea of division with remainders. So that's coming up.

Now on another topic, suppose we had to find all the positive factors of 36. Now, a test question may ask how many factors does 36 have? There are also procedures, for example, when we get to factoring quadratics where it's gonna be important to find the factors of a number. Also, when we find the prime factorization, it will help to find some factors of the number.

But suppose we had to find all the factors. Well it's certainly clear, for example, we know 1 is a factor, 36 is a factor, maybe it's clear that 2,3, and 4 are factors, 7 is not a factor. We could kind of go haphazardly, but there's actually a system we can follow to find every factor. The easiest way is to list the factor pairs, the pairs of numbers that have a product of 36.

So the first pair I'm gonna list is 1 and 36. Those are clearly two factors, 1 is a factor of every number 36 has to be a factor of itself. Well 2 goes into 36, so that would be 2 times 18, that gives me two more factors. 3 goes into 36, that would give me 3 times 12. 4 goes into 36, that would be 4 times 9.

5 does not go into 36, so we'll skip that. And then we get to 6, well, 6 times itself, so we stop with six. Once you get to a number, you get to a number times itself, or you pass, and you reconnect with one of the numbers you have earlier in the list, something like that, then you know you're done. So now we know we are guaranteed that we have found every single factor of 36.

So, we don't count 6 twice. Notice that 36 has nine positive factors including 1 and itself. And so the procedure, again, you just list them in pairs. You skip the numbers. If any number, if the number's not divisible by that particular number, you just skip it.

And then you keep going until either you double back on yourself or you come to a number that's the product of itself. For small numbers, numbers less than 100, we can count the positive factors simply by making a list of factor pairs. We'll learn another, more efficient technique for larger numbers later in this module.

So for example, the test could give you a number like 12,600, and ask you to find, how many factors does this have? And it would just take far too much time to list all those factors. It would be, it would take like ten minutes to make a factor list. You wouldn't have time to do it on the test. Later on in this module we'll actually talk about an incredibly efficient procedure and we'll actually use that particular number, 12,600.

We'll find the number of factors in it very easily, so don't worry about that now. Just know that for numbers less than 100, you can just make the list of factor pairs. Finally, a note about negative integers. I've been focusing mostly on positive integers here.

Technically it is true that both +4 and -4 are factors of -12. Both +4 and -4 are divisors of -12. And -12 is divisible by both +4 and -4. It's true that every positive factor, for example 36 had nine positive factors, we could just put a negative sign in front of them. There'd be nine negative factors also.

So usually we're not as concerned with the negative factors, cuz they're just repeats of the positive factors. And that's why the test almost never asks about that or expects you to know it. But technically, factors and divisors could be negative. And that is something that we may have to watch out for on a more advanced question. In summary, we talked about the interrelated ideas of factors, divisors, and divisibility.

We talked about options for division that don't come out even, remainders versus non-integer quotients. Remember, remainders we'll be discussing in much greater detail a few modules from now. We talked about listing the factors of a number by listing factor pairs and introduced the idea of counting factors.

For larger numbers, we'll have another video later on. We'll talk about that in greater detail. And we talked about the seldom-used rules for negative factors.