How to list all categories in three divs dynamically based on the quatity of categories?

You can use get_categories() function.

<?php
  $cats =  get_categories();
  $count = 15;  // Total Categorie
  $linetotal=round($count/3); ?>  // Part of Categorie

  <div class="category_list">
    <ul><?php
      $totalcount = 1;
      $totalpart = 1;
      foreach($cats as $cat): ?>
        <li><?php echo $cat->name; ?></li><?php
        if($totalcount % $linetotal == 0 && $totalpart < 3) { 
          echo '</ul></div><div class="category_list"><ul>';
              $totalpart++;
        }
        $totalcount++;
      endforeach;?>
    </ul>
  </div>