Transcript
整数属性，这是一个测试最喜欢的问题之一，询问关于整数属性。
所以这个模块我们将谈论整数的所有属性，当然整数是非分数号。
它们包括所有正数，正整数1,2,3。它们还包括0和负整数，-1，-2，-3，所有这些。
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. 事实上，我们在该模块中讨论的大部分内容仅涉及正整数。 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. 在这个特殊的视频中，我们将讨论一些相互关联的想法，这些是您肯定需要了解和理解测试的数学术语。 这些是测试问题中出现的术语。所以三个要了解，首先是因素，除数和可分离。
让我们首先用公式讨论这些术语和一些简单的数字示例。 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. 然后我们说A和B是C的因素。换句话说，我们可以将它们乘以将C获取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.
另请注意，每个整数都是本身的一个因素。因此，每个大于1的正整数至少有两个因素，1和本身。 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 84的一个因素必须是84.如果c / a = b，换句话说，我们可以执行这个司和 该部门的输出仍然是一个整数，然后我们说A是C的除法，因为它均匀地分成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. 所以我们在乘法的上下文中使用一个，还有一个在划分的背景下，但真的，它意味着完全相同。
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是24的除数，24个是可被8的。所有三个都说完全相同的东西。
它们沟通完全相同的数学信息。所以不要困惑， 测试可能在不同的问题中使用不同的措辞。类似地，8不是12,8的因子不是12的除数，而12是 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. 如果两个数字均匀划分，那么（整数）除以（整数）等于（整数）：商是一个整数，就是这样。 这很容易。但如果两个项目不均匀分割，我们有两种可能的选择， each perfectly correct.
选项编号是具有整数商和整数余数。所以我们会说12次进入8人， a quotient of 1, with a remainder of 4. Option number two would be to express the quotient as a fraction or decimal. 所以我们可以说12除以8，该分数简化了3岁以上。我们也可以将其写成混合数字，一半，和 我们可以把它写为1.5。
So both of these are perfectly correct. You will need to understand how to employ either option, 但你永远不必决定使用哪一个。测试问题的性质将始终明确哪一个 他们的意思是。换句话说，他们是否会谈论剩余者或是否 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.
在此模块的后期，我们将在更大程度的复杂性讨论剩余者。 所以,现在只是一个味道。We actually have a later video that are gonna be devoted exclusively to the idea of division with remainders. So that's coming up.
现在在另一个主题上，假设我们必须找到36的所有积极因素。现在，一个测试问题可能会问36有多少因素？ 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. 此外，当我们找到原始分解时，它将有助于找到数量的一些因素。
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, 也许很明显，2,3和4是因素，7不是一个因素。我们可能会偶然，但是 there's actually a system we can follow to find every factor. The easiest way is to list the factor pairs, 具有36个产品的数量。
So the first pair I'm gonna list is 1 and 36. Those are clearly two factors, 图1是每个数字36的一个因素必须是本身的一个因素。嗯2进入36，所以这将是2次，给了我两个的因素。 3进入36，这将给我3次12.4进入36，这将是4倍。
5没有进入36，所以我们会跳过这一点。然后我们到了6岁，本身6次，所以我们停下来六。 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, 这样的东西，那么你知道你已经完成了。所以现在我们知道我们保证我们已经找到了每一个单一因素36。
所以，我们不计数6两次。请注意，36有九个阳性因素，包括1和本身。 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.
例如，测试可以给你一个数字12,600，并要求你找到，这有多少因素？ 它需要太多时间来列出所有这些因素。这将是，它需要十分钟来制作一个因素清单。 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.
我们会非常容易地找到它的因素数量，所以现在不要担心。 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.
技术上，+4和-4都是-12的因素。+4和-4都是-12的除数。 And -12 is divisible by both +4 and -4. It's true that every positive factor, for example 36 had nine positive factors, 我们可以在他们面前放一个负面的标志。也有九个消极因素。
因此，我们通常不会与消极因素那么关注，因为他们只是重复积极因素。 这就是为什么测试几乎从未问过或期望您知道它。但从技术上讲，因素和除数可能是消极的。 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, 和可分性。
We talked about options for division that don't come out even, remainders versus non-integer quotients. 请记住，我们将在现在的一些模块中讨论余额。 我们通过上市因子对列出了一个数字的因素，并引入了计数因子的想法。
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 transcript
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. 事实上，我们在该模块中讨论的大部分内容仅涉及正整数。 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. 在这个特殊的视频中，我们将讨论一些相互关联的想法，这些是您肯定需要了解和理解测试的数学术语。 这些是测试问题中出现的术语。所以三个要了解，首先是因素，除数和可分离。
让我们首先用公式讨论这些术语和一些简单的数字示例。 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. 然后我们说A和B是C的因素。换句话说，我们可以将它们乘以将C获取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.
另请注意，每个整数都是本身的一个因素。因此，每个大于1的正整数至少有两个因素，1和本身。 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 84的一个因素必须是84.如果c / a = b，换句话说，我们可以执行这个司和 该部门的输出仍然是一个整数，然后我们说A是C的除法，因为它均匀地分成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. 所以我们在乘法的上下文中使用一个，还有一个在划分的背景下，但真的，它意味着完全相同。
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是24的除数，24个是可被8的。所有三个都说完全相同的东西。
它们沟通完全相同的数学信息。所以不要困惑， 测试可能在不同的问题中使用不同的措辞。类似地，8不是12,8的因子不是12的除数，而12是 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. 如果两个数字均匀划分，那么（整数）除以（整数）等于（整数）：商是一个整数，就是这样。 这很容易。但如果两个项目不均匀分割，我们有两种可能的选择， each perfectly correct.
选项编号是具有整数商和整数余数。所以我们会说12次进入8人， a quotient of 1, with a remainder of 4. Option number two would be to express the quotient as a fraction or decimal. 所以我们可以说12除以8，该分数简化了3岁以上。我们也可以将其写成混合数字，一半，和 我们可以把它写为1.5。
So both of these are perfectly correct. You will need to understand how to employ either option, 但你永远不必决定使用哪一个。测试问题的性质将始终明确哪一个 他们的意思是。换句话说，他们是否会谈论剩余者或是否 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.
在此模块的后期，我们将在更大程度的复杂性讨论剩余者。 所以,现在只是一个味道。We actually have a later video that are gonna be devoted exclusively to the idea of division with remainders. So that's coming up.
现在在另一个主题上，假设我们必须找到36的所有积极因素。现在，一个测试问题可能会问36有多少因素？ 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. 此外，当我们找到原始分解时，它将有助于找到数量的一些因素。
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, 也许很明显，2,3和4是因素，7不是一个因素。我们可能会偶然，但是 there's actually a system we can follow to find every factor. The easiest way is to list the factor pairs, 具有36个产品的数量。
So the first pair I'm gonna list is 1 and 36. Those are clearly two factors, 图1是每个数字36的一个因素必须是本身的一个因素。嗯2进入36，所以这将是2次，给了我两个的因素。 3进入36，这将给我3次12.4进入36，这将是4倍。
5没有进入36，所以我们会跳过这一点。然后我们到了6岁，本身6次，所以我们停下来六。 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, 这样的东西，那么你知道你已经完成了。所以现在我们知道我们保证我们已经找到了每一个单一因素36。
所以，我们不计数6两次。请注意，36有九个阳性因素，包括1和本身。 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.
例如，测试可以给你一个数字12,600，并要求你找到，这有多少因素？ 它需要太多时间来列出所有这些因素。这将是，它需要十分钟来制作一个因素清单。 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.
我们会非常容易地找到它的因素数量，所以现在不要担心。 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.
技术上，+4和-4都是-12的因素。+4和-4都是-12的除数。 And -12 is divisible by both +4 and -4. It's true that every positive factor, for example 36 had nine positive factors, 我们可以在他们面前放一个负面的标志。也有九个消极因素。
因此，我们通常不会与消极因素那么关注，因为他们只是重复积极因素。 这就是为什么测试几乎从未问过或期望您知道它。但从技术上讲，因素和除数可能是消极的。 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, 和可分性。
We talked about options for division that don't come out even, remainders versus non-integer quotients. 请记住，我们将在现在的一些模块中讨论余额。 我们通过上市因子对列出了一个数字的因素，并引入了计数因子的想法。
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.
