Math 4: Recognizing prime and composite numbers

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1Determine whether the following numbers are prime, composite or neither.

1請確定以下各數字為質數、合數、或兩者皆非。

2So just as a bit of review, a prime number is a natural number, so one of the counting numbers 1, 2, 3, 4, 5, 6, so on and so forth - that has exactly two factors.

2稍稍回顧一下，質數是自然數，是計數數字 1、2、3、4、5、6... 等等之一個數字，而它恰恰有兩個因數。

3So its factors are 1 and itself.

3即其因數為 1 及它本身。

4So an example of a prime is 3.

4所以 3 是質數的一個例子。

5There's only two natural numbers that are divisible into 3 - 1 and 3.

5只有兩個自然數可以整除 3 即 1 和 3。

6Or another way to think about it is, the only way to get 3 as a product of other natural numbers is 1 times 3.

6或者換一個方式思考，將 3 作為其他自然數的乘積之唯一方法是 1 乘以 3。

7So it only has 1 and itself.

7因此它只有 1 和 它自己。

8A composite number is a natural number that has more than just 1 and itself as factors.

8一個合數是一個自然數，它不僅僅具有 1 和它本身為其因數。

9And we'll see examples of that and neither -- we’ll see an interesting case of that in this problem.

9我們將看到那些(合數)的例子，以及兩者皆非--我們將在本題目中看到一個有趣的例子。

10So first let's think about 24.

10好，首先讓我們考慮 24。

11So let’s think about all of the -- I guess you could think of it as the natural numbers or the whole numbers, although 0 is also included in whole numbers.

11因此，讓我們考慮所有 -- 我猜您可能將其 (24) 代表所有自然數或整數，儘管整數中也包含 0。

12Let’s think of all of the natural counting numbers that we can actually divide into 24 without having any remainder.

12讓我們考慮所有那些我們可以用來整除 24 而沒有餘數的自然數。

13We’d consider those the factors.

13我們要評估它們是否為因數。

14Well, clearly it is divisible by 1 and 24.

14好吧！顯然它(24)可以被 1 和 24 整除。

15In fact, 1 times 24 is equal to 24.

15事實上，1 乘以 24 等於 24。

16But it's also divisible by 2.

16而它(24)也可以被 2 整除。

172 times 12 is 24.

172 乘以 12 得 24。

18So it’s also divisible by 12.

18因此它(24)也可以被 12 整除。

19And it is also divisible by 3.

19而且它(24)也可以被 3 整除。

203 times 8 is also equal to 24.

203 乘以 8 也等於 24。

21And even at this point, we don't actually have to find all of the factors to realize it's not prime.

21而甚至在此時，我們實際上並不需要找出所有因數來確定它(24)不是質數。

22It clearly has more factors than just 1 and itself.

22它顯然有 1 和 24 之外更多的因數。

23So then it is clearly going to be composite.

23因此它(24)顯然將會是一個合數。

24This is going to be composite.

24這將會是合數。

25Now, let’s just finish factoring it just since we started it.

25現在，既然我們已經開始了，就讓我們完成它(24)的因數分解。

26It’s also divisible by 4.

26它(24)也可以被 4 整除。

27And 4 times 6 -- had just enough space to do that.

27還有 4 乘以 6 -- 正好有足够的空間可以做這個。

284 times 6 is also 24.

284 乘以 6 也是 24。

29So these are all of the factors of 24, clearly more than just 1 in 24.

29因此這些都是 24 的因數，顯然的不只是 1 和 24。

30Now let’s think about 2.

30現在讓我們來看看 2。

31Well, the non-zero whole numbers that are divisible into 2, well, 1 times 2 definitely works, 1 and 2.

31是的，可以整除 2 的非零整數，是的，1 乘以 2 必定可以，1 和 2。

32But there really aren't any others that are divisible into 2.

32可是真的沒有其他的數可以整除 2。

33And so it only has two factors, 1 and itself, and that's the definition of a prime number.

33因此它(2)只有兩個因數，即 1 和自己，而這就是質數的定義。

34So 2 is prime.

34所以 2 是質數。

35And 2 is interesting because it is the only even prime number.

35另外， 2 很有意思，因為它(2)是唯一的偶質數。

36And that might be common sense to you.

36而這可能對你而言是常識。

37Because by definition, an even number is divisible by 2.

37因為根據定義，偶數就是可以被 2 整除的數。

38So 2 is clearly divisible by 2.

38所以 2 顯然的可以被 2 整除。

39That's what makes it even.

39這使得它(2)成為偶數。

40But it’s only divisible by 2 and 1.

40而它(2)也只能被 2 和 1 整除。

41So that's what makes it prime.

41這使得它(2)因而成為質數。

42But anything that’s even is going to be divisible by 1, itself and 2.

42另外，任何的偶數都可以被 1，它自己，和 2 整除。

43Any other number that is even is going to be visible by 1, itself, and 2.

43任何其他的偶數都可以被 1，它自己，和 2 整除。

44So by definition, it’s going to have 1 and itself and something else.

44所以根據定義，它(任何其他的偶數)將有 1，它自己和其他數(為因數)。

45So it’s going to be composite.

45因此它(任何其他的偶數)將會是合數。

46So 2 is prime.

46故 2 是質數。

47Every other even number other than 2 is composite.

47其他不是 2 的偶數都是合數。

48Now, here is an interesting case.

48再來，這兒有一個有趣的例子。

491--1 is only divisible by 1.

491-- 1 只能被 1 整除。

50So it is not prime, technically, because it only has 1 as a factor.

50所以它(1)不是質數，技術上來說，因為它(1)只有一個因數。

51It does not have two factors.

51它(1)沒有兩個因數。

521 is itself.

521 是它自己。

53But in order to be prime, you have to have exactly two factors.

53但要成為質數，你必須要恰有兩個因數。

541 has only one factor.

541 只有一個因數。

55In order to be composite you have to have more than two factors.

55但要成為合數，你必須要有兩個以上的因數。

56You have to have 1, yourself, and some other numbers.

56你必須要有 1，你自己，和一些其他的數。

57So it's not composite.

57因此它(1)不是合數。

58So 1 is neither prime nor composite.

58所以 1 既不是質數也不是合數。

59And then finally we get to 17.

59接下來終於我們來到 17。

6017 is divisible by 1 and 17.

6017 可以被 1 和 17 整除。

61It’s not divisible by 2, not divisible by 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, or 16.

61它(17)可能被 2 整除，不能被 3,4…, 16 整除。

62So it has exactly two factors -- 1 and itself.

62所以它(17)恰有兩個因數--即 1 和它自己。

63So 17 is once again -- 17 is prime.

63故 17 又是 -- 17 是質數。