如何求两个数的最小公倍数

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共同创作者是 Grace Imson, MA

参考

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倍数是一个数乘以整数得到的结果。一组数字的最小公倍数（简称为LCM）是这组数共有倍数中最小的一个数。要找出最小公倍数，你需要先确定各个数字的因数。求解最小公倍数的方法有很多。本文介绍的方法适用于求两个和更多数字的最小公倍数。

方法 1
方法 1 的 4:

列出数字的所有倍数

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    这个方法最适用于计算两个小于10的数字的公倍数，如果你面对的是比较大或比较多的数字，最好使用其它方法。
    • 例如，我们需要找到5和8的最小公倍数。由于这两个数字都比较小，适合使用这个方法求出它们的最小公倍数。
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    用第一个数字乘以不同的整数就能得到它的倍数。 [1] 也就是说，你可以直接查看乘法表，找到一个数的倍数。
    • 例如，第一个数字5的倍数有5、10、15、20、25、30、35和40。
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    用相同的整数乘以第二个数字，得到几个倍数，来和之前的一组倍数进行比较。
    • 在我们的示例中，数字8的倍数有8、16、24、32、40、48、56和64。
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    你可能需要列出更多倍数，来找到相同的那个倍数。你能找到的最小的相同数字就是最小公倍数。 [2]
    • 例如，5和8的倍数里都有40，而且它是最小的相同倍数，所以40是5和8的最小公倍数。
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方法 2
方法 2 的 4:

使用素因式分解法

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    这个方法最适用于计算两个大于10的数字的公倍数，如果你面对的是比较小的数字，最好使用其它方法快速求出最小公倍数。
    • 例如，如果你要找出数字20和84的最小公倍数，你可以使用这种方法。
  2. {"smallUrl":"https:\/\/www.wikihow.com\/images_en\/thumb\/a\/af\/Find-the-Least-Common-Multiple-of-Two-Numbers-Step-6-Version-4.jpg\/v4-460px-Find-the-Least-Common-Multiple-of-Two-Numbers-Step-6-Version-4.jpg","bigUrl":"https:\/\/www.wikihow.com\/images\/thumb\/a\/af\/Find-the-Least-Common-Multiple-of-Two-Numbers-Step-6-Version-4.jpg\/v4-728px-Find-the-Least-Common-Multiple-of-Two-Numbers-Step-6-Version-4.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"}
    你可以将第一个数字因式分解成它的素数因数，得到的几个素数因数相乘，就能够得到原始数字。你可以画出因子树来将数字分解成素数。完成因式分解后，重新写出等式。等式的一边是被分解的数字，另一边是素数因数相乘。
    • 例如， ， ，因此，20的素数因数有2、2、和5。重新写出等式，得到 。
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    用相同的方式分解第二个数字，找到它的素数因数，各个素数因数相乘能够得到第二个数字。
    • 例如， , , 以及 。因此，84的素数因数有2、7、3和2。 重新写出等式，得到 。
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    在你写下每个因数的同时，请在因式分解的等式中划掉对应的数值。
    • 例如，两个数字拥有共同的因数2，因此，写下因数 ，并将每个因式中的2划掉。
    • 两个数字还拥有另一个2作为共同的因数，因此，再写下第二个数字2，并写成两数相乘： ，然后划掉因式分解式子里的另一个2。
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    剩余的因数是指划掉公因数后，几个因式分解的等式中没有被划掉的因数。也就是两个数字的因数中不相同的那些。 [3]
    • 例如，在等式 中，两个2是两个数字共同的因数，因此你会划掉两个2。还剩下一个5，将5添加到上面的乘法式子中，得到： 。
    • 在等式 中，你也划掉了两个2，还剩下了7和3，将这两个数字也加到乘式中，变成： 。
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    将上面写下的所有因数相乘，得到最小公倍数。
    • 在我们的例子中， 。因此，20和84的最小公倍数是420。
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方法 3
方法 3 的 4:

使用网格法或梯形法

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    井字形的网格由两组平行线交叉组成，两组平行线彼此相互垂直，形成三行三列的网格，看上去像是手机或键盘上的井字键（#）。在网格最上方中央的方格内写下你的第一个数字，在网格右上角的方格内写下第二个数字。 [4]
    • 例如，如果你想找到数字18和30的最小公倍数，请将18写在最上方中央的方格内，在网格右上角的方格写下30。
  2. {"smallUrl":"https:\/\/www.wikihow.com\/images_en\/thumb\/b\/b5\/Find-the-Least-Common-Multiple-of-Two-Numbers-Step-12-Version-4.jpg\/v4-460px-Find-the-Least-Common-Multiple-of-Two-Numbers-Step-12-Version-4.jpg","bigUrl":"https:\/\/www.wikihow.com\/images\/thumb\/b\/b5\/Find-the-Least-Common-Multiple-of-Two-Numbers-Step-12-Version-4.jpg\/v4-728px-Find-the-Least-Common-Multiple-of-Two-Numbers-Step-12-Version-4.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"}
    将这个数字写在网格左上角的方格内。最好使用素数因数，这会大大方便后续的计算，但是也不是必须的。
    • 在求解18和30的最小公倍数例题中，由于18和30都是偶数，所以都能整除2，将2写在网格左上角的方格内。
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    将除得的商写在每个数字下面的方格中。进行除法计算就能得到商。
    • 例如， ，在数字18下面写下9。
    • ，在网格中30下面的格子里写下15。
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    如果两个商没有公因数，可以跳过这一步直接进入下一步。如果它们有公因数，请写在网格中央偏左的格子里。
    • 例如，9和15的公因数为3，所以将3写在网格中央偏左的格子里。
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    将结果写在上一步结果的下面。
    • 例如， ，将3写在9下方的方格内。
    • ，将5写在15下方的方格内。
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    然后按照上面的步骤计算除法，直到两个商没有相同的因数为止。
  7. {"smallUrl":"https:\/\/www.wikihow.com\/images_en\/thumb\/9\/9f\/Find-the-Least-Common-Multiple-of-Two-Numbers-Step-17.jpg\/v4-460px-Find-the-Least-Common-Multiple-of-Two-Numbers-Step-17.jpg","bigUrl":"https:\/\/www.wikihow.com\/images\/thumb\/9\/9f\/Find-the-Least-Common-Multiple-of-Two-Numbers-Step-17.jpg\/v4-728px-Find-the-Least-Common-Multiple-of-Two-Numbers-Step-17.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"}
    圆圈连起来，就像是画出了一个大写的“L”字母。将圈出的所有数字相乘。 [5]
    • 在我们的例题中，2和3位于网格的第一列，3和5位于网格的最后一行，写出数学式： 。
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    将所有因数相乘，得到的结果就是原来两数的最小公倍数。 [6]
    • 例如： 。因此，18和30的最小公倍数是90。
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方法 4
方法 4 的 4:

使用欧几里德算法

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    “被除数”是除法运算中被另一个数所除的数；“除数”是被除数除以的数字；“商”是除法的最后结果；“余数”是整数被整除以后余下的数字。 [7]
    • 例如，在方程 :
      15 是被除数
      6 是除数
      2 是商
      3 是余数。
  2. {"smallUrl":"https:\/\/www.wikihow.com\/images_en\/thumb\/9\/9a\/Find-the-Least-Common-Multiple-of-Two-Numbers-Step-20.jpg\/v4-460px-Find-the-Least-Common-Multiple-of-Two-Numbers-Step-20.jpg","bigUrl":"https:\/\/www.wikihow.com\/images\/thumb\/9\/9a\/Find-the-Least-Common-Multiple-of-Two-Numbers-Step-20.jpg\/v4-728px-Find-the-Least-Common-Multiple-of-Two-Numbers-Step-20.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"}
    公式是 被除数 = 除数 × 商 + 余数 。 [8] 你需要用这个公式，根据欧几里得算法求出两个数字的最大公约数。
    • 例如， 。
    • 最大公约数是两个数字公有的最大除数或因子。 [9]
    • 使用本方法，你需要先求出最大公约数，然后通过它来找到最小公倍数。
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    建立两个数字的“商-余数”方程。
    • 例如，如果你要求210和45的最小公倍数，那么方程的形式是 。
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    建立两个数字的“商-余数”方程。
    • 例如， 。
  5. {"smallUrl":"https:\/\/www.wikihow.com\/images_en\/thumb\/6\/6d\/Find-the-Least-Common-Multiple-of-Two-Numbers-Step-23.jpg\/v4-460px-Find-the-Least-Common-Multiple-of-Two-Numbers-Step-23.jpg","bigUrl":"https:\/\/www.wikihow.com\/images\/thumb\/6\/6d\/Find-the-Least-Common-Multiple-of-Two-Numbers-Step-23.jpg\/v4-728px-Find-the-Least-Common-Multiple-of-Two-Numbers-Step-23.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"}
    每一个新方程中，你都需要使用原除数作为新的被除数，使用余数作为新的除数。 [10]
    • 例如， 。因为，最后的余数是0，所以你不需要再继续除下去了。
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    这个数字就是两个数字的最大公约数。 [11]
    • 例如，因为最后一个方程 中，除数是15，所以15就是210和45的最大公约数。
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    用它们的乘积除以它们的最大公约数。最后的结果就是两个数字的最小公倍数。 [12]
    • 例如， 。用乘积除以最大公约数，得到 。所以，630就是210和45的最小公倍数。
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小提示

  • 如果你需要求多个数字的最小公倍数，那么上述的方法需要稍作更改。例如，要找到16、20和32的最小公倍数，请先使用上述方法求出16和20的最小公倍数（80）。再求出80和32的最小公倍数，最后计算结果是160。
  • 最小公倍数有很多用途。最常见的用途是，当你计算分数的加减法时，几个分数的分母数字必须是相同的；如果分母不同，你需要将分子和分母同时乘以一个数，使得几个分数的分母变成相同的数字。最好的办法就是求出最小公分母（LCD），也就是分母的最小公倍数（LCM）。
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参考

  1. ↑ https://www.mathsisfun.com/definitions/multiple.html
  2. ↑ https://www.mathsisfun.com/least-common-multiple.html
  3. ↑ https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-lcm/v/least-common-multiple-exercise
  4. ↑ https://www.youtube.com/watch?v=NKlOm3tPQiw
  5. ↑ https://www.youtube.com/watch?v=b6qehkDuioQ
  6. ↑ https://www.youtube.com/watch?v=NKlOm3tPQiw
  7. ↑ http://www.ducksters.com/kidsmath/division_basics.php
  8. ↑ https://people.richland.edu/james/lecture/m116/polynomials/zeros.html
  9. ↑ http://mathworld.wolfram.com/GreatestCommonDivisor.html
  1. ↑ https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/the-euclidean-algorithm
  2. ↑ https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/the-euclidean-algorithm
  3. ↑ http://csharphelper.com/blog/2014/08/calculate-the-greatest-common-divisor-gcd-and-least-common-multiple-lcm-of-two-integers-in-c/

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