阶乘的最低有效非零数字

Question

我正在尝试计算阶乘中的最低有效非零数字。

我有以下片段：

$(document).ready(function() {
  $('#submit').click(function() {
    var n = $('#number').val();
    get_result(n);
  });
});

function get_result(n) {
  var factorial = 1;
  var factorial2 = 1;
  for (i = 1; i <= n; i++) {
    factorial = factorial * i;
  }
  var count_5 = 0;
  for (j = 1; j <= n; j++) {
    if (j % 5 != 0) {
      factorial2 = factorial2 * (j % 10);
      factorial2 = factorial2 % 10;
    } else if (j % 5 == 0) {
      count_5 = 1;
    }
  }
  if (count_5 == 1) {
    factorial2 = factorial2 * 5;
  }
  console.log(factorial2);
  factorial2 = factorial2.toString();
  var digit = 0;
  for (i = 0; i < factorial2.length; i++) {
    if (factorial2[i] != '0') {
      digit = factorial2[i];
    }
  }
  $('#display').text("Factorial of " + n + " is " + factorial);
  $('#display2').text("Least significant digit of Factorial of " + n + " is " + digit);
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div id="display">

</div>
<div id="display2">

</div>
<input type="text" value="" id="number">
<input type="submit" id="submit">

作为上述代码的一部分，为了计算最低有效非零数字，我首先忽略所有 5 的倍数，其次，在阶乘计算的每一步，我从10 为了在computation.Eventually的每一步只保留非零数字，我将factorial2的最终值乘以5然后将其转换为字符串并找到非零的最后一次出现- 字符串中的数字为零。

以上代码似乎适用于 n=1,2.......,8 的值。但是在 n=9 时，代码 returns 最低有效非零数字为 3 而它应该返回 8.

For example : Factorial(9) = 362880 , hence least significant non-zero digit = 8.

可能是什么错误，我应该如何纠正它？还有另一种性能更好的方法来计算这个结果吗？

Note : I've included the code to calculate the factorial just for verification purposes, my ultimate aim is to just calculate the least significant non-zero digit and not the factorial for a worst possible case when n is a billion(when actually computing and reading the factorial is not feasible or advisable).

Answer 1

问题是 5 不会简单地消失。它们与 2 结合生成 0。因此，在 5 的倍数（如 15 或 35）或具有很多 2 的幂（如 24）的数字之后，你会遇到问题。最好的办法可能是记录 2 的个数，并在每个 5 的倍数时减少它（2 总是多于 5）。（此外，一旦您费尽心思找到了不带 0 的数字，就无需将其转换为字符串。）

$(document).ready(function() {
  $('#submit').click(function() {
    var n = $('#number').val();
    get_result(n);
  });
});

function get_result(n) {
  var factorial = 1;
  var factorial2 = 1;
  for ( var i = 1; i <= n; i++ ) {
    factorial = factorial * i;
  }
  var extra2s = 0;
  for ( var j = 1; j <= n; j++ ) {
    var jcopy = j;
    while( jcopy%10 == 0 ) {
      jcopy /= 10;
    }
    while( jcopy%2==0 ) {
      extra2s++;
      jcopy /= 2;
    }
    while( jcopy%5==0 ) {
      extra2s--;
      jcopy /= 5;
    }
    jcopy %= 10;
    factorial2 = (factorial2 * jcopy)%10;
  }
  for ( var k = 0 ; k < extra2s ; k++ ) {
    factorial2 = (factorial2 * 2)%10;
  }
  var digit = factorial2;
  $('#display').text("Factorial of " + n + " is " + factorial);
  $('#display2').text("Least significant digit of Factorial of " + n + " is " + digit);
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div id="display">

</div>
<div id="display2">

</div>
<input type="text" value="" id="number">
<input type="submit" id="submit">

阶乘的最低有效非零数字

Least significant non-zero digit of a factorial

javascript

jquery

html

performance

factorial