5.4质因数分解

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  质因数分解

  

  [Figure 1]

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  Connor is working on prime factorization for his math homework. He needs to find the  prime  numbers that, when multiplied together, produce the number 82. How can Connor complete this problem?
  ::Connor正在研究数学功课的质因数。 他需要找到质数,当合并后产生82号。 Connor如何解决这个问题?

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  In this concept, you will learn to write the prime factorization of given numbers using a  factor tree .
  ::在此概念中,您将学会使用因子树写出给定数字的质因子化。

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  Prime Factorization Using Factor Trees 
  ::使用因数树

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  When a number is  factored , it is broken down into two  factors  that are either prime numbers or composite numbers.  Prime numbers  are numbers that have only two factors, one and itself, and  composite numbers  are numbers that have more than two factors. Some examples of prime numbers are 2, 3, 11, etc.  Prime factorization  is the process of breaking down a number into a product of all prime numbers.
  ::当一个数字被乘以系数时,它被分为两个因素,要么是质数,要么是复合数字。先数是只有两个因素的数字,一个是本身,而复合数字是有两个以上因素的数字。一些质数的例子有2、3、11等。先入为主的分化是将一个数分解成所有质数的产物的过程。

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  Here is a composite number.
  ::这是一个复合数字。

  36

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  The number 36 can be factored several different ways, but let’s factor it with  $\begin{array}{r}6\times 6\end{array}$ .
  ::数字36可以用几种不同的方式来计算,但用 6 × 6 来计算。

  $$\begin{array}{r}36=6\times 6\end{array}$$

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  These two factors are not prime factors. Therefore, both factors can be factored again.
  ::这两个因素不是首要因素,因此,两个因素都可以再次考虑。

  $$\begin{array}{r}36=6\times 6=2\times 3\times 2\times 3\end{array}$$

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  2 and 3 are both prime numbers.
  ::2和3都是质数。

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  One way to organize the factors is using a  factor tree .
  ::一种组织因素的方法是使用因素树。

  $$\begin{array}{r}\begin{array}{rcl}& & 36\\ & & /\backslash \\ & & 6\times 6\\ & & /\backslash /\backslash \\ & & 2\times 32\times 3\\ \\ & & 36=2\times 2\times 3\times 3\end{array}\end{array}$$

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  The number is written at the top of the factor tree. Then it is broken down into a factor pair,  $\begin{array}{r}6\times 6\end{array}$ . 6 can further be factored so the factor pairs are written underneath the 6. Each number is continued to be factored until the factors are all prime numbers. Note that 36 is written as a product of its primes at the bottom of the factor tree. Write the 2s together and the 3s together. Grouping like factors will help keep track of them.
  ::数字写在系数树的顶部。 然后将数字细分为因数对, 6× 6。 6 可以进一步计算, 使因数对写在6 下。 每个因数都继续计算, 直到因数都是质数。 请注意, 36 是作为因数树底部的质数的产物写成的。 将 2 和 3 一起写在一起。 类似因素的分组有助于跟踪它们 。

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  The prime factorization is written using  exponential notation , a method of writing repeated multiplication.
  ::基因子化是用指数符号写成的,这是写出重复乘法的一种方法。

  $$\begin{array}{r}36=2\times 2\times 3\times 3=2^2\times 3^2\end{array}$$

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  The base is the number being repeated and the exponent is the number of times the number is being multiplied. 2 times 2 is written as base 2 with the exponent 2, the number of times 2 is multiplied by itself. 3 times 3 is written as base 3 with an exponent of 2.
  ::基数是重复的数字,引号是数字乘以的倍数。 2乘2是基数2,2乘2是基数2,2乘本身乘以2。3乘3是基数3,引号为2。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Connor’s prime factorization math problem.
  ::更早之前,你曾遇到康纳的 质因数计算问题。

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  Connor needs to find the prime factorization of 82. Use a factor tree to solve this problem.
  ::Connor需要找到82的因数 使用因数树解决这个问题。

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  First, start with 82 at the top of the factor tree.
  ::首先,从系数树顶部的82开始。

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  Then, begin by factoring 82 using factor pairs.
  ::然后,首先将82乘以因子对数乘以82。

  $$\begin{array}{r}\begin{array}{rcl}& & 82\\ & & /\backslash \\ & & 2\times 41\\ \\ & & 82=2\times 41\end{array}\end{array}$$

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  Connor finds that 82 is the product of only 2 prime numbers, 2 and 41.
  ::Connor发现82是只有2个质数,2和41的产物。

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  Example 2
  ::例2

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  Write the prime factorization of 25.
  ::写入25的基数乘数

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  First, start with 25 at the top of your factor tree.
  ::首先,从你因子树顶部的25个开始

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  Then, factor 25 into the product of all prime numbers. 25 can be factored into 5 times 5. 5 is a prime number so this tells you that you have reached the bottom of the factor tree.
  ::然后,在所有质数的产物中,乘以25。25可以乘以5乘以5,5是质数,这表示你到达了因数树底部。

  $$\begin{array}{r}\begin{array}{rcl}& & 25\\ & & /\backslash \\ & & 5\times 5\\ \\ & & 25=5\times 5\end{array}\end{array}$$

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  Next, write the factors in exponential notation. The base is 5 and the exponent is 2.
  ::接下来,在指数符号中写入系数。基数为 5,引号为 2。

  $$\begin{array}{r}25=5^2\end{array}$$

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  The prime factorization of 25 is  $\begin{array}{r}{5}^2\end{array}$ .
  ::25的基数是5 2

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  Example 3
  ::例3

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  Write the prime factorization for the following number.
  ::写入以下数字的基数乘数。

  $$\begin{array}{r}48\end{array}$$

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  First, start with 48 at the top of your factor tree.
  ::首先,从你因子树顶部的48开始

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  Then, factor 48 into the product of all prime numbers.
  ::然后,所有质数的产物中的系数48。

  $$\begin{array}{r}\begin{array}{rcl}& & 48\\ & & /\backslash \\ & & 4\times 12\\ & & /\backslash /\backslash \\ & & 2\times 22\times 6\\ & & /\backslash \\ & & 2\times 3\\ \\ & & 48=2\times 2\times 2\times 2\times 3\end{array}\end{array}$$

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  Next, write the factors in exponential notation.
  ::接下来,在指数符号中写入系数。

  $$\begin{array}{r}48=2^4\times 3\end{array}$$

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  The prime factorization of 48 is  $\begin{array}{r}{2}^4\times 3\end{array}$ .
  ::48的基数为2 4x 3。

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  Example 4
  ::例4

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  Write the prime factorization for the following number.
  ::写入以下数字的基数乘数。

  $$\begin{array}{r}100\end{array}$$

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  First, start with 100 at the top of your factor tree.
  ::首先,从你因子树顶部的100开始。

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  Then, factor 100 into the product of all prime numbers.
  ::然后,将100乘以所有质数的产物。

  $$\begin{array}{r}\begin{array}{rcl}& & 100\\ & & /\backslash \\ & & 2\times 50\\ & & /\backslash \\ & & 2\times 25\\ & & /\backslash \\ & & 5\times 5\\ \\ & & 100=2\times 2\times 5\times 5\end{array}\end{array}$$

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  Next, write the factors in exponential notation.
  ::接下来,在指数符号中写入系数。

  $$\begin{array}{r}100=2^2\times 5^2\end{array}$$

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  The prime factorization of 100 is  $\begin{array}{r}{2}^2\times 5^2\end{array}$ .
  ::基数100是 2 2 × 5 2 。

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  Example 5
  ::例5

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  Write the prime factorization for the following number.
  ::写入以下数字的基数乘数。

  $$\begin{array}{r}144\end{array}$$

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  First, start with 144 at the top of your factor tree.
  ::首先,从你因子树顶端的144个开始

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  Then, factor 144 into the product of all prime numbers.
  ::然后,所有质数的产物乘以144。

  $$\begin{array}{r}\begin{array}{rcl}& & 144\\ & & /\backslash \\ & & 12\times 12\\ & & /\backslash /\backslash \\ & & 2\times 62\times 6\\ & & /\backslash /\backslash \\ & & 2\times 32\times 3\\ \\ & & 144=2\times 2\times 2\times 2\times 3\times 3\end{array}\end{array}$$

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  Next, write the factors in exponential notation.
  ::接下来,在指数符号中写入系数。

  $$\begin{array}{r}144=2^4\times 3^2\end{array}$$

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  The prime factorization of 144 is  $\begin{array}{r}{2}^4\times 3^2\end{array}$ .
  ::144的基因数为2 4x3 2。

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  Review
  ::回顾

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  Write the prime factorization of each number using exponential notation.
  ::使用指数符号书写每个数字的基数乘数。

  1. 56
  2. 14
  3. 121
  4. 84
  5. 50
  6. 64
  7. 72
  8. 16
  9. 24
  10. 300
  11. 128
  12. 312
  13. 525
  14. 169
  15. 213

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  Review (Answers) 
  ::回顾(答复)

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  Click   to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

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  Resources
  ::资源