數與座標系/2 因數與倍數-範例

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  • 範例6
    • 311,313,317,319,323 中何者為質數？
    • 3600
      1. 正因數有幾個，其總和若干，其乘積若干
      2. 正因數為36之倍數有幾個，其總和若干
      3. 正因數中為完全平方者有幾個
    • 範例6類題1
      • 已知：
      • 求解：
    • 範例6類題2
      • 已知：
      • 求解：

範例6

題目

  • 311,313,317,319,323 中何者為質數？
  • 3600
    1. 正因數有幾個，其總和若干，其乘積若干
    2. 正因數為36之倍數有幾個，其總和若干
    3. 正因數中為完全平方者有幾個

答案

  • 311,313,317為質數

詳解

1

  1. 311,313,317,319,323 中何者為質數？
    1. \sqrt{323}<19^{2}=361，小於19之質數有17,13,11,7,5,3,2，其中11,5,3,2可用倍數判定方法檢驗是否為已知數之因數，而17,13,7則用除法檢驗之。
    2. 經倍數判定方法知11,5,3,2中除11可整除319外，其餘皆不可整除已知數。
    3. 用除法檢驗17,13,7是否為319以外的已知數如下：
      • \left\{ \begin{align} & \frac{311}{17}=18+\frac{5}{17} \\ & \frac{311}{13}=24+\frac{9}{13} \\ & \frac{311}{7}=44+\frac{3}{7} \\ \end{align} \right.\Rightarrow \left\{ \begin{align} & \frac{313}{17}=18+\frac{7}{17} \\ & \frac{313}{13}=24+\frac{11}{13} \\ & \frac{313}{7}=44+\frac{5}{7} \\ \end{align} \right.\Rightarrow \left\{ \begin{align} & \frac{317}{17}=18+\frac{11}{17} \\ & \frac{317}{13}=24+\frac{15}{13} \\ & \frac{317}{7}=44+\frac{9}{7} \\ \end{align} \right.\Rightarrow \left\{ \begin{align} & \frac{323}{17}=18+{\color{Red}\frac{17}{17}} \\ & \frac{323}{13}=24+\frac{21}{13} \\ & \frac{323}{7}=44+\frac{15}{7} \\ \end{align} \right.
      • 由檢驗式中得知只有17為323之因數，故質數為311,313,317\#。
  2. 3600
    1. 3600=36\times 100=6^{2}\times 10^{2}=\left( 2\times 3 \right)^{2}\times \left( 2\times 5 \right)^{2}=2^{2}\times 3^{2}\times 2^{2}\times 5^{2}=2^{4}\times 3^{2}\times 5^{2}
    2. 故3600之正因數表列如下：
      • {\color{Red}\left\{ \begin{align} & {\color{Red}\left\{ \begin{align} & f_{1,1,1}=2^{0}\times 3^{0}\times 5^{0} \\ & f_{2,1,1}=2^{1}\times 3^{0}\times 5^{0} \\ & f_{3,1,1}=2^{2}\times 3^{0}\times 5^{0} \\ & f_{4,1,1}=2^{3}\times 3^{0}\times 5^{0} \\ & f_{5,1,1}=2^{4}\times 3^{0}\times 5^{0} \\ \end{align} \right.} \\ & {\color{OliveGreen}\left\{ \begin{align} & f_{1,2,1}=2^{0}\times 3^{1}\times 5^{0} \\ & f_{2,2,1}=2^{1}\times 3^{1}\times 5^{0} \\ & f_{3,2,1}=2^{2}\times 3^{1}\times 5^{0} \\ & f_{4,2,1}=2^{3}\times 3^{1}\times 5^{0} \\ & f_{5,2,1}=2^{4}\times 3^{1}\times 5^{0} \\ \end{align} \right.} \\ & {\color{Blue}\left\{ \begin{align} & f_{1,3,1}=2^{0}\times 3^{2}\times 5^{0} \\ & f_{2,3,1}=2^{1}\times 3^{2}\times 5^{0} \\ & f_{3,3,1}=2^{2}\times 3^{2}\times 5^{0} \\ & f_{4,3,1}=2^{3}\times 3^{2}\times 5^{0} \\ & f_{5,3,1}=2^{4}\times 3^{2}\times 5^{0} \\ \end{align} \right.} \\ \end{align} \right.}
        {\color{OliveGreen}\left\{ \begin{matrix} {\color{Red}\left\{ \begin{align} & f_{1,1,2}=2^{0}\times 3^{0}\times 5^{1} \\ & f_{2,1,2}=2^{1}\times 3^{0}\times 5^{1} \\ & f_{3,1,2}=2^{2}\times 3^{0}\times 5^{1} \\ & f_{4,1,2}=2^{3}\times 3^{0}\times 5^{1} \\ & f_{5,1,2}=2^{4}\times 3^{0}\times 5^{1} \\ \end{align} \right. } \\ {\color{OliveGreen}\left\{ \begin{align} & f_{1,2,2}=2^{0}\times 3^{1}\times 5^{1} \\ & f_{2,2,2}=2^{1}\times 3^{1}\times 5^{1} \\ & f_{3,2,2}=2^{2}\times 3^{1}\times 5^{1} \\ & f_{4,2,2}=2^{3}\times 3^{1}\times 5^{1} \\ & f_{5,2,2}=2^{4}\times 3^{1}\times 5^{1} \\ \end{align} \right. } \\ {\color{Blue}\left\{ \begin{align} & f_{1,3,2}=2^{0}\times 3^{2}\times 5^{1} \\ & f_{2,3,2}=2^{1}\times 3^{2}\times 5^{1} \\ & f_{3,3,2}=2^{2}\times 3^{2}\times 5^{1} \\ & f_{4,3,2}=2^{3}\times 3^{2}\times 5^{1} \\ & f_{5,3,2}=2^{4}\times 3^{2}\times 5^{1} \\ \end{align} \right.} \\ \end{matrix} \right.}
        {\color{Blue}\left\{ \begin{matrix} {\color{Red}\left\{ \begin{align} & f_{1,1,3}=2^{0}\times 3^{0}\times 5^{2} \\ & f_{2,1,3}=2^{1}\times 3^{0}\times 5^{2} \\ & f_{3,1,3}=2^{2}\times 3^{0}\times 5^{2} \\ & f_{4,1,3}=2^{3}\times 3^{0}\times 5^{2} \\ & f_{5,1,3}=2^{4}\times 3^{0}\times 5^{2} \\ \end{align} \right.} \\ {\color{OliveGreen}\left\{ \begin{align} & f_{1,2,3}=2^{0}\times 3^{1}\times 5^{2} \\ & f_{2,2,3}=2^{1}\times 3^{1}\times 5^{2} \\ & f_{3,2,3}=2^{2}\times 3^{1}\times 5^{2} \\ & f_{4,2,3}=2^{3}\times 3^{1}\times 5^{2} \\ & f_{5,2,3}=2^{4}\times 3^{1}\times 5^{2} \\ \end{align} \right.} \\ {\color{Blue}\left\{ \begin{align} & f_{1,3,3}=2^{0}\times 3^{2}\times 5^{2} \\ & f_{2,3,3}=2^{1}\times 3^{2}\times 5^{2} \\ & f_{3,3,3}=2^{2}\times 3^{2}\times 5^{2} \\ & f_{4,3,3}=2^{3}\times 3^{2}\times 5^{2} \\ & f_{5,3,3}=2^{4}\times 3^{2}\times 5^{2} \\ \end{align} \right.} \\ \end{matrix} \right.}
    3. 3600之正因數個數為：\underbrace{\left( 1+4 \right)}_{2^{0},2^{1},2^{2},2^{3},2^{4}}\times \underbrace{\left( 1+2 \right)}_{3^{0},3^{1},3^{2}}\times \underbrace{\left( 1+2 \right)}_{5^{0},5^{1},5^{2}}=45\#

2

  • 3600之正因數總和為：\underbrace{\left( 2^{0}\times 3^{0}\times 5^{0} \right)}_{f_{1,1,1}}+\underbrace{\left( 2^{1}\times 3^{0}\times 5^{0} \right)}_{f_{2,1,1}}+\cdots +\underbrace{\left( 2^{4}\times 3^{2}\times 5^{2} \right)}_{f_{5,3,3}}
    =\left( 2^{0}+2^{1}+2^{2}+2^{3}+2^{4} \right)\times \left( 3^{0}+3^{1}+3^{2} \right)\times \left( 5^{0}+5^{1}+5^{2} \right)
    =31\times 13\times 31=12493\#

3

  1. \left\{ \begin{align} & P=\underbrace{\left( 2^{0}\times 3^{0}\times 5^{0} \right)}_{f_{1,1,1}}\times \underbrace{\left( 2^{1}\times 3^{0}\times 5^{0} \right)}_{f_{2,1,1}}\times \cdots \times \underbrace{\left( 2^{3}\times 3^{2}\times 5^{2} \right)}_{f_{4,3,3}}\times \underbrace{\left( 2^{4}\times 3^{2}\times 5^{2} \right)}_{f_{5,3,3}} \\ & P=\underbrace{\left( 2^{4}\times 3^{2}\times 5^{2} \right)}_{f_{5,3,3}}\times \underbrace{\left( 2^{3}\times 3^{2}\times 5^{2} \right)}_{f_{4,3,3}}\times \cdots \times \underbrace{\left( 2^{1}\times 3^{0}\times 5^{0} \right)}_{f_{2,1,1}}\times \underbrace{\left( 2^{0}\times 3^{0}\times 5^{0} \right)}_{f_{1,1,1}} \\ \end{align} \right.
  2. P^{2}=\left( 2^{4}\times 3^{2}\times 5^{2} \right)\times \left( 2^{4}\times 3^{2}\times 5^{2} \right)\times \cdots \times \left( 2^{4}\times 3^{2}\times 5^{2} \right)\times \left( 2^{4}\times 3^{2}\times 5^{2} \right)=\left( 2^{4}\times 3^{2}\times 5^{2} \right)^{45}
  3. P=\left( 2^{4}\times 3^{2}\times 5^{2} \right)^{\frac{45}{2}}=\left( 3600 \right)^{\frac{45}{2}}=\left( 60^{2} \right)^{\frac{45}{2}}=60^{45}\#

4

  1. 3600之正因數中為36倍數的因數標示紅色如下表：
    • \left\{ \begin{align} & \left\{ \begin{align} & f_{1,1,1}=2^{0}\times 3^{0}\times 5^{0} \\ & f_{2,1,1}=2^{1}\times 3^{0}\times 5^{0} \\ & f_{3,1,1}=2^{2}\times 3^{0}\times 5^{0} \\ & f_{4,1,1}=2^{3}\times 3^{0}\times 5^{0} \\ & f_{5,1,1}=2^{4}\times 3^{0}\times 5^{0} \\ \end{align} \right. \\ & \left\{ \begin{align} & f_{1,2,1}=2^{0}\times 3^{1}\times 5^{0} \\ & f_{2,2,1}=2^{1}\times 3^{1}\times 5^{0} \\ & f_{3,2,1}=2^{2}\times 3^{1}\times 5^{0} \\ & f_{4,2,1}=2^{3}\times 3^{1}\times 5^{0} \\ & f_{5,2,1}=2^{4}\times 3^{1}\times 5^{0} \\ \end{align} \right. \\ & \left\{ \begin{align} & f_{1,3,1}=2^{0}\times 3^{2}\times 5^{0} \\ & f_{2,3,1}=2^{1}\times 3^{2}\times 5^{0} \\ & f_{3,3,1}=2^{2}\times 3^{2}\times 5^{0}={\color{Red}36\times 2^{0}\times 5^{0}} \\ & f_{4,3,1}=2^{3}\times 3^{2}\times 5^{0}={\color{Red}36\times 2^{1}\times 5^{0}} \\ & f_{5,3,1}=2^{4}\times 3^{2}\times 5^{0}={\color{Red}36\times 2^{2}\times 5^{0}} \\ \end{align} \right. \\ \end{align} \right.
      \left\{ \begin{matrix} \left\{ \begin{align} & f_{1,1,2}=2^{0}\times 3^{0}\times 5^{1} \\ & f_{2,1,2}=2^{1}\times 3^{0}\times 5^{1} \\ & f_{3,1,2}=2^{2}\times 3^{0}\times 5^{1} \\ & f_{4,1,2}=2^{3}\times 3^{0}\times 5^{1} \\ & f_{5,1,2}=2^{4}\times 3^{0}\times 5^{1} \\ \end{align} \right. \\ \left\{ \begin{align} & f_{1,2,2}=2^{0}\times 3^{1}\times 5^{1} \\ & f_{2,2,2}=2^{1}\times 3^{1}\times 5^{1} \\ & f_{3,2,2}=2^{2}\times 3^{1}\times 5^{1} \\ & f_{4,2,2}=2^{3}\times 3^{1}\times 5^{1} \\ & f_{5,2,2}=2^{4}\times 3^{1}\times 5^{1} \\ \end{align} \right. \\ \left\{ \begin{align} & f_{1,3,2}=2^{0}\times 3^{2}\times 5^{1} \\ & f_{2,3,2}=2^{1}\times 3^{2}\times 5^{1} \\ & f_{3,3,2}=2^{2}\times 3^{2}\times 5^{1}={\color{Red}36\times 2^{0}\times 5^{1}} \\ & f_{4,3,2}=2^{3}\times 3^{2}\times 5^{1}={\color{Red}36\times 2^{1}\times 5^{1}} \\ & f_{5,3,2}=2^{4}\times 3^{2}\times 5^{1}={\color{Red}36\times 2^{2}\times 5^{1}} \\ \end{align} \right. \\ \end{matrix} \right.
      \left\{ \begin{matrix} \left\{ \begin{align} & f_{1,1,3}=2^{0}\times 3^{0}\times 5^{2} \\ & f_{2,1,3}=2^{1}\times 3^{0}\times 5^{2} \\ & f_{3,1,3}=2^{2}\times 3^{0}\times 5^{2} \\ & f_{4,1,3}=2^{3}\times 3^{0}\times 5^{2} \\ & f_{5,1,3}=2^{4}\times 3^{0}\times 5^{2} \\ \end{align} \right. \\ \left\{ \begin{align} & f_{1,2,3}=2^{0}\times 3^{1}\times 5^{2} \\ & f_{2,2,3}=2^{1}\times 3^{1}\times 5^{2} \\ & f_{3,2,3}=2^{2}\times 3^{1}\times 5^{2} \\ & f_{4,2,3}=2^{3}\times 3^{1}\times 5^{2} \\ & f_{5,2,3}=2^{4}\times 3^{1}\times 5^{2} \\ \end{align} \right. \\ \left\{ \begin{align} & f_{1,3,3}=2^{0}\times 3^{2}\times 5^{2} \\ & f_{2,3,3}=2^{1}\times 3^{2}\times 5^{2} \\ & f_{3,3,3}=2^{2}\times 3^{2}\times 5^{2}={\color{Red}36\times 2^{0}\times 5^{2}} \\ & f_{4,3,3}=2^{3}\times 3^{2}\times 5^{2}={\color{Red}36\times 2^{1}\times 5^{2}} \\ & f_{5,3,3}=2^{4}\times 3^{2}\times 5^{2}={\color{Red}36\times 2^{2}\times 5^{2}} \\ \end{align} \right. \\ \end{matrix} \right.
  2. 36\times m=2^{4}\times 3^{2}\times 5^{2}=36\times 2^{2}\times 5^{2}\Rightarrow m=2^{2}\times 5^{2}
  3. 故正因數中為36之倍數有：\underbrace{\left( 1+2 \right)}_{2^{0},2^{1},2^{2}}\times \underbrace{\left( 1+2 \right)}_{5^{0},5^{1},5^{2}}=9\#
  4. 故正因數中為36之倍數之總和為：36\times \left( 2^{0}+2^{1}+2^{2} \right)\times \left( 5^{0}+5^{1}+5^{2} \right)=36\times 7\times 31=7812\#

5

  1. 3600之正因數中為完全平方者標示紅色如下表：
    • \left\{ \begin{align} & \left\{ \begin{align} & f_{1,1,1}={\color{Red}2^{0}\times 3^{0}\times 5^{0}} \\ & f_{2,1,1}=2^{1}\times 3^{0}\times 5^{0} \\ & f_{3,1,1}={\color{Red}2^{2}\times 3^{0}\times 5^{0}} \\ & f_{4,1,1}=2^{3}\times 3^{0}\times 5^{0} \\ & f_{5,1,1}={\color{Red}2^{4}\times 3^{0}\times 5^{0}} \\ \end{align} \right. \\ & \left\{ \begin{align} & f_{1,2,1}=2^{0}\times 3^{1}\times 5^{0} \\ & f_{2,2,1}=2^{1}\times 3^{1}\times 5^{0} \\ & f_{3,2,1}=2^{2}\times 3^{1}\times 5^{0} \\ & f_{4,2,1}=2^{3}\times 3^{1}\times 5^{0} \\ & f_{5,2,1}=2^{4}\times 3^{1}\times 5^{0} \\ \end{align} \right. \\ & \left\{ \begin{align} & f_{1,3,1}={\color{Red}2^{0}\times 3^{2}\times 5^{0}} \\ & f_{2,3,1}=2^{1}\times 3^{2}\times 5^{0} \\ & f_{3,3,1}={\color{Red}2^{2}\times 3^{2}\times 5^{0}} \\ & f_{4,3,1}=2^{3}\times 3^{2}\times 5^{0} \\ & f_{5,3,1}={\color{Red}2^{4}\times 3^{2}\times 5^{0}} \\ \end{align} \right. \\ \end{align} \right.
      \left\{ \begin{matrix} \left\{ \begin{align} & f_{1,1,2}=2^{0}\times 3^{0}\times 5^{1} \\ & f_{2,1,2}=2^{1}\times 3^{0}\times 5^{1} \\ & f_{3,1,2}=2^{2}\times 3^{0}\times 5^{1} \\ & f_{4,1,2}=2^{3}\times 3^{0}\times 5^{1} \\ & f_{5,1,2}=2^{4}\times 3^{0}\times 5^{1} \\ \end{align} \right. \\ \left\{ \begin{align} & f_{1,2,2}=2^{0}\times 3^{1}\times 5^{1} \\ & f_{2,2,2}=2^{1}\times 3^{1}\times 5^{1} \\ & f_{3,2,2}=2^{2}\times 3^{1}\times 5^{1} \\ & f_{4,2,2}=2^{3}\times 3^{1}\times 5^{1} \\ & f_{5,2,2}=2^{4}\times 3^{1}\times 5^{1} \\ \end{align} \right. \\ \left\{ \begin{align} & f_{1,3,2}=2^{0}\times 3^{2}\times 5^{1} \\ & f_{2,3,2}=2^{1}\times 3^{2}\times 5^{1} \\ & f_{3,3,2}=2^{2}\times 3^{2}\times 5^{1} \\ & f_{4,3,2}=2^{3}\times 3^{2}\times 5^{1} \\ & f_{5,3,2}=2^{4}\times 3^{2}\times 5^{1} \\ \end{align} \right. \\ \end{matrix} \right.
      \left\{ \begin{matrix} \left\{ \begin{align} & f_{1,1,3}={\color{Red}2^{0}\times 3^{0}\times 5^{2}} \\ & f_{2,1,3}=2^{1}\times 3^{0}\times 5^{2} \\ & f_{3,1,3}={\color{Red}2^{2}\times 3^{0}\times 5^{2}} \\ & f_{4,1,3}=2^{3}\times 3^{0}\times 5^{2} \\ & f_{5,1,3}={\color{Red}2^{4}\times 3^{0}\times 5^{2}} \\ \end{align} \right. \\ \left\{ \begin{align} & f_{1,2,3}=2^{0}\times 3^{1}\times 5^{2} \\ & f_{2,2,3}=2^{1}\times 3^{1}\times 5^{2} \\ & f_{3,2,3}=2^{2}\times 3^{1}\times 5^{2} \\ & f_{4,2,3}=2^{3}\times 3^{1}\times 5^{2} \\ & f_{5,2,3}=2^{4}\times 3^{1}\times 5^{2} \\ \end{align} \right. \\ \left\{ \begin{align} & f_{1,3,3}={\color{Red}2^{0}\times 3^{2}\times 5^{2}} \\ & f_{2,3,3}=2^{1}\times 3^{2}\times 5^{2} \\ & f_{3,3,3}={\color{Red}2^{2}\times 3^{2}\times 5^{2}} \\ & f_{4,3,3}=2^{3}\times 3^{2}\times 5^{2} \\ & f_{5,3,3}={\color{Red}2^{4}\times 3^{2}\times 5^{2}} \\ \end{align} \right. \\ \end{matrix} \right.
  2. 3600=4^{2}\times 9^{1}\times 25^{1}，其中4,9,25為完全平方。
  3. 正因數中為完全平方者有：\underbrace{\left( 1+2 \right)}_{4^{0},4^{1},4^{2}}\times \underbrace{\left( 1+1 \right)}_{9^{0},9^{1}}\times \underbrace{\left( 1+1 \right)}_{25^{0},25^{1}}=12\#

範例6類題1

已知

求解

答案

詳解

範例6類題2

已知

求解

答案

詳解


範例7

已知

  • 七位數23ab421為99的倍數

求解

  • 數對 \left( a,b \right)

答案

  • \left( a,b \right)=\left( 2,4 \right)

詳解

  1. 因為a,b\,皆為個位數，所以\left\{ \begin{align} & 0\le a\le 9 \\ & 0\le b\le 9 \\ \end{align} \right.\Rightarrow \left\{ \begin{align} & 0\le a+b\le 18 \\ & -9\le a-b\le 9 \\ \end{align} \right.
  2. 99|23ab421\Rightarrow \left\{ \begin{align} & 9|23ab421 \\ & 11|23ab421 \\ \end{align} \right.\Rightarrow \left\{ \begin{align} & 9|\left( 2+3+a+b+4+2+1 \right) \\ & 11|\left[ \left( 2+a+4+1 \right)-\left( 3+b+2 \right) \right] \\ \end{align} \right.，提示：9與11之倍數判定
    \Rightarrow \left\{ \begin{align} & 9|\left( 12+a+b \right) \\ & 11|\left( 2+a-b \right) \\ \end{align} \right.\Rightarrow \left\{ \begin{align} & \left( 12+a+b \right)=18,27,... \\ & \left( 2+a-b \right)=0,11,... \\ \end{align} \right.
    \Rightarrow \left\{ \begin{align} & \left( a+b \right)=\left( 18-12 \right)or\left( 27-12 \right)=6or15 \\ & \left( a-b \right)=\left( 0-2 \right)or\left( 11-2 \right)=-2or9 \\ \end{align} \right.
    \Rightarrow \left\{ \begin{matrix} \left\{ \begin{matrix} a+b=6 \\ a-b=-2 \\ \end{matrix}\Rightarrow \left\{ \begin{matrix} a=2 \\ b=4 \\ \end{matrix}\# \right. \right. \\ \left\{ \begin{matrix} a+b=6 \\ a-b=9 \\ \end{matrix}\Rightarrow \left\{ \begin{matrix} a=\frac{15}{2} \\ b=-\frac{3}{2} \\ \end{matrix}\left( N/A \right) \right. \right. \\ \left\{ \begin{matrix} a+b=15 \\ a-b=-2 \\ \end{matrix}\Rightarrow \left\{ \begin{matrix} a=\frac{13}{2} \\ b=\frac{17}{2} \\ \end{matrix}\left( N/A \right) \right. \right. \\ \left\{ \begin{matrix} a+b=15 \\ a-b=9 \\ \end{matrix}\Rightarrow \left\{ \begin{matrix} a=12 \\ b=3 \\ \end{matrix}\left( N/A \right) \right. \right. \\ \end{matrix} \right.

範例7類題1

已知

求解

答案

詳解

範例7類題2

已知

求解

答案

詳解

取自"http://www.come2pass.com/wiki/index.php/%E6%95%B8%E8%88%87%E5%BA%A7%E6%A8%99%E7%B3%BB/2_%E5%9B%A0%E6%95%B8%E8%88%87%E5%80%8D%E6%95%B8-%E7%AF%84%E4%BE%8B"