阶乘因式分解（一）

╰半夏微凉° 2022-06-10 12:56 218阅读 0赞

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      <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px"> 
       <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px"> 
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       <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; font-family:Verdana; line-height:21px; text-align:left"> 
        <h2 style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; font-size:18px; font-weight:normal; color:rgb(113,32,21); font-family:微软雅黑,黑体; text-align:center"> 阶乘因式分解（一）</h2> 
        <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; text-align:center">
          时间限制： 
         <span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(113,32,21)">3000</span>&nbsp;ms &nbsp;|&nbsp; 内存限制： 
         <span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(113,32,21)">65535</span>&nbsp;KB 
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          难度： 
         <span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(113,32,21)">2</span> 
        </div> 
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         &nbsp; 
       </div> 
       <dl style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; font-family:Verdana; line-height:21px; text-align:left">
         描述 
        <dd style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px"> 
         <p style="margin-top:1em; margin-bottom:1em; padding-top:0px; padding-bottom:0px"> <span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; font-size:12px"><span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px"><span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(76,51,229)"><span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(255,255,0)"><span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px"><span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; background-color:rgb(0,0,0)"><span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px"><span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px"><span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(0,0,0); background-color:rgb(255,255,255)">给定两个数m,n,其中m是一个素数。</span></span></span></span></span></span></span></span></span></p> 
         <p style="margin-top:1em; margin-bottom:1em; padding-top:0px; padding-bottom:0px"> <span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; font-size:12px"><span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px">将n（0&lt;=n&lt;=10000）的阶乘分解质因数，求其中有多少个m。</span></span></p> 
         <p style="margin-top:1em; margin-bottom:1em; padding-top:0px; padding-bottom:0px"> <span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; line-height:25px; color:rgb(78,78,78); font-family:Arial,Helvetica,simsun,u5b8bu4f53; background-color:rgb(173,224,255)"></span></p> 
         <p style="margin-top:0px; margin-bottom:10px; padding-top:0px; padding-bottom:0px; line-height:25px; display:inline!important"> <span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(78,78,78); font-family:Arial,Helvetica,simsun,u5b8bu4f53; background-color:rgb(173,224,255)"><span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; line-height:23px; font-size:12px"><span style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; background-color:rgb(255,255,255); color:rgb(0,0,0)"><span style="font-size:12px">注：^为求幂符号。</span></span></span></span></p> 
         <p style="margin-top:1em; margin-bottom:1em; padding-top:0px; padding-bottom:0px"> </p> 
        </dd> 
       </dl> 
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         &nbsp; 
       </div> 
       <dl style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; font-family:Verdana; line-height:21px; text-align:left">
         输入 
        <dd style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px">
          第一行是一个整数s（0&lt;s&lt;=100)，表示测试数据的组数 
         <br> 随后的s行, 每行有两个整数n，m。 
        </dd> 输出 
        <dd style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px">
          输出m的个数。 
        </dd> 样例输入 
        <dd style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px"> 
         <pre style="margin-top:0px; margin-bottom:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; white-space:pre-wrap; word-wrap:break-word; line-height:14px">2100 516 2</pre> 
        </dd> 样例输出 
        <dd style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px"> 
         <pre style="margin-top:0px; margin-bottom:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; white-space:pre-wrap; word-wrap:break-word; line-height:14px">2415
解题：这道题主要是减少算法的时间复杂度，用数学方法解决。如果球出阶乘再做因式分解，一定会超时，而且工作量大，不可取。现在有一种方法
可以解决这一问题。将m除以n的值保留然后再让该值除以n如此往复，并将每次求得的值相加就得到想要的结果。南工上还有一道题与此类似，唉！忘了……</pre> 
         <pre style="margin-top:0px; margin-bottom:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; white-space:pre-wrap; word-wrap:break-word; line-height:14px"><p style="margin-top:5px; margin-right:auto; margin-bottom:5px; margin-left:auto; padding-top:0px; padding-bottom:0px; border-top-width:0px; border-right-width:0px; border-bottom-width:0px; border-left-width:0px; border-style:initial; border-color:initial; list-style-type:none; list-style-position:initial; word-wrap:normal; word-break:normal; line-height:24px; white-space:normal; color:rgb(64,50,38); font-family:verdana,'ms song',宋体,Arial,微软雅黑,Helvetica,sans-serif">给定两个数m,n</p><p style="margin-top:5px; margin-right:auto; margin-bottom:5px; margin-left:auto; padding-top:0px; padding-bottom:0px; border-top-width:0px; border-right-width:0px; border-bottom-width:0px; border-left-width:0px; border-style:initial; border-color:initial; list-style-type:none; list-style-position:initial; word-wrap:normal; word-break:normal; line-height:24px; white-space:normal; color:rgb(64,50,38); font-family:verdana,'ms song',宋体,Arial,微软雅黑,Helvetica,sans-serif">求n!分解质因数后因子m的个数。</p><p style="margin-top:5px; margin-right:auto; margin-bottom:5px; margin-left:auto; padding-top:0px; padding-bottom:0px; border-top-width:0px; border-right-width:0px; border-bottom-width:0px; border-left-width:0px; border-style:initial; border-color:initial; list-style-type:none; list-style-position:initial; word-wrap:normal; word-break:normal; line-height:24px; white-space:normal; color:rgb(64,50,38); font-family:verdana,'ms song',宋体,Arial,微软雅黑,Helvetica,sans-serif">这道题涉及到了大数问题，如果相乘直接求的话会超出数据类型的范围。</p><p style="margin-top:5px; margin-right:auto; margin-bottom:5px; margin-left:auto; padding-top:0px; padding-bottom:0px; border-top-width:0px; border-right-width:0px; border-bottom-width:0px; border-left-width:0px; border-style:initial; border-color:initial; list-style-type:none; list-style-position:initial; word-wrap:normal; word-break:normal; line-height:24px; white-space:normal; color:rgb(64,50,38); font-family:verdana,'ms song',宋体,Arial,微软雅黑,Helvetica,sans-serif">下面给出一种效率比较高的算法，我们一步一步来。</p><p style="margin-top:5px; margin-right:auto; margin-bottom:5px; margin-left:auto; padding-top:0px; padding-bottom:0px; border-top-width:0px; border-right-width:0px; border-bottom-width:0px; border-left-width:0px; border-style:initial; border-color:initial; list-style-type:none; list-style-position:initial; word-wrap:normal; word-break:normal; line-height:24px; white-space:normal; color:rgb(64,50,38); font-family:verdana,'ms song',宋体,Arial,微软雅黑,Helvetica,sans-serif">n!=1*2*3*……*(m-2)*(m-1)*m</p><p style="margin-top:5px; margin-right:auto; margin-bottom:5px; margin-left:auto; padding-top:0px; padding-bottom:0px; border-top-width:0px; border-right-width:0px; border-bottom-width:0px; border-left-width:0px; border-style:initial; border-color:initial; list-style-type:none; list-style-position:initial; word-wrap:normal; word-break:normal; line-height:24px; white-space:normal; color:rgb(64,50,38); font-family:verdana,'ms song',宋体,Arial,微软雅黑,Helvetica,sans-serif">可以表示成所有和m倍数有关的乘积再乘以其他和n没有关系的</p><p style="margin-top:5px; margin-right:auto; margin-bottom:5px; margin-left:auto; padding-top:0px; padding-bottom:0px; border-top-width:0px; border-right-width:0px; border-bottom-width:0px; border-left-width:0px; border-style:initial; border-color:initial; list-style-type:none; list-style-position:initial; word-wrap:normal; word-break:normal; line-height:24px; white-space:normal; color:rgb(64,50,38); font-family:verdana,'ms song',宋体,Arial,微软雅黑,Helvetica,sans-serif">&nbsp;&nbsp;&nbsp;&nbsp;=(m*2m*3m*......*km)*ohter&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;other是不含n因子的数的乘积&nbsp;&nbsp;&nbsp;因为 km&lt;=n而k肯定是最大值&nbsp;&nbsp;所以k=n/m</p><p style="margin-top:5px; margin-right:auto; margin-bottom:5px; margin-left:auto; padding-top:0px; padding-bottom:0px; border-top-width:0px; border-right-width:0px; border-bottom-width:0px; border-left-width:0px; border-style:initial; border-color:initial; list-style-type:none; list-style-position:initial; word-wrap:normal; word-break:normal; line-height:24px; white-space:normal; color:rgb(64,50,38); font-family:verdana,'ms song',宋体,Arial,微软雅黑,Helvetica,sans-serif">&nbsp;&nbsp;&nbsp;&nbsp;=m^k*(1*2*......*k)*other&nbsp;&nbsp;</p><p style="margin-top:5px; margin-right:auto; margin-bottom:5px; margin-left:auto; padding-top:0px; padding-bottom:0px; border-top-width:0px; border-right-width:0px; border-bottom-width:0px; border-left-width:0px; border-style:initial; border-color:initial; list-style-type:none; list-style-position:initial; word-wrap:normal; word-break:normal; line-height:24px; white-space:normal; color:rgb(64,50,38); font-family:verdana,'ms song',宋体,Arial,微软雅黑,Helvetica,sans-serif">&nbsp;&nbsp;&nbsp;&nbsp;=m^k*k!*other　　　　　</p><p style="margin-top:5px; margin-right:auto; margin-bottom:5px; margin-left:auto; padding-top:0px; padding-bottom:0px; border-top-width:0px; border-right-width:0px; border-bottom-width:0px; border-left-width:0px; border-style:initial; border-color:initial; list-style-type:none; list-style-position:initial; word-wrap:normal; word-break:normal; line-height:24px; white-space:normal; color:rgb(64,50,38); font-family:verdana,'ms song',宋体,Arial,微软雅黑,Helvetica,sans-serif">从这个表达式中可以提取出k个m，然后按照相同的方法循环下去可以求出k!中因子m的个数。</p><p style="margin-top:5px; margin-right:auto; margin-bottom:5px; margin-left:auto; padding-top:0px; padding-bottom:0px; border-top-width:0px; border-right-width:0px; border-bottom-width:0px; border-left-width:0px; border-style:initial; border-color:initial; list-style-type:none; list-style-position:initial; word-wrap:normal; word-break:normal; line-height:24px; white-space:normal; color:rgb(64,50,38); font-family:verdana,'ms song',宋体,Arial,微软雅黑,Helvetica,sans-serif">每次求出m的个数的和就是n!中因子m的总个数。</p></pre> #include&lt;stdio.h&gt; 
         <br> int main() 
         <br> { 
         <br> &nbsp;&nbsp;&nbsp; int N,m,n,sum; 
         <br> &nbsp;&nbsp;&nbsp; scanf("%d",&amp;N); 
         <br> &nbsp;&nbsp;&nbsp; while(N--) 
         <br> &nbsp;&nbsp;&nbsp; { 
         <br> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; sum = 0; 
         <br> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; scanf("%d%d",&amp;m,&amp;n); 
         <br> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; while(m) 
         <br> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; { 
         <br> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; sum = sum + m/n; 
         <br> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; m = m/n; 
         <br> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; } 
         <br> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; printf("%d\n",sum); 
         <br> &nbsp;&nbsp;&nbsp; } 
         <br> &nbsp;&nbsp;&nbsp;&nbsp; 
         <br> &nbsp;&nbsp;&nbsp; return 0; 
         <br> 
        </dd> 
        <dd style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px">
          } 
        </dd> 
        <dd style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px">
          其他解法 
        </dd> 
        <dd style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px"> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           #include&lt;stdio.h&gt; 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           int main() 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           { 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           &nbsp; &nbsp;int i,j,k,m,n,s; 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           &nbsp; &nbsp;scanf("%d",&amp;s); 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           &nbsp; &nbsp;while(s--) 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           &nbsp; &nbsp;{ 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           &nbsp; &nbsp;scanf("%d%d",&amp;n,&amp;m); 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           &nbsp; &nbsp;&nbsp; &nbsp;for(i=m,j=0;i&lt;=n;i++) 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           &nbsp; &nbsp;&nbsp; &nbsp;for(k=i;!(k%m);j++)//j不能写在这里，否则只是n有几个m的个数j。 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           &nbsp; &nbsp;&nbsp; &nbsp;k=k/m; 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           &nbsp; &nbsp;&nbsp; &nbsp;printf("%d\n",j); 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           &nbsp; &nbsp;} 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           &nbsp; &nbsp;return 0; 
         </div> 
         <div style="margin-top:0px; margin-right:0px; margin-bottom:0px; margin-left:0px; padding-top:0px; padding-right:0px; padding-bottom:0px; padding-left:0px; color:rgb(73,73,73); font-family:simsun; background-color:rgb(236,240,252)">
           } 
         </div> 
        </dd> 
       </dl> 
      </div> 
     </div> 
    </div> </td> 
  </tr> 
 </tbody> 
</table>