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Теория: Умножение больших натуральных чисел

Задание

Найти произведение чисел:

    \(\displaystyle 4\)\(\displaystyle 5\)\(\displaystyle 9\)\(\displaystyle 7\)\(\displaystyle 4\)
   x     
      \(\displaystyle 5\)\(\displaystyle 4\)\(\displaystyle 8\)

   
\(\displaystyle +\)        
   
\(\displaystyle +\)        
   

 

 

Решение

1. Первое действие.

Умножаем единицы числа \(\displaystyle 54{\underline 8}\) (это число \(\displaystyle 8\)) на число \(\displaystyle 45974{\small:}\)

 

\(\displaystyle 45974 \cdot 8=367792{\small.}\)

 

    \(\displaystyle 4\)\(\displaystyle 5\)\(\displaystyle 9\)\(\displaystyle 7\)\(\displaystyle 4\)
   x     
      \(\displaystyle 5\)\(\displaystyle 4\)\(\displaystyle \bf8\)

   \(\displaystyle \bf3\)\(\displaystyle \bf6\)\(\displaystyle \bf7\)\(\displaystyle \bf7\)\(\displaystyle \bf9\)\(\displaystyle \bf2\)
\(\displaystyle +\)        
  \(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\) 
\(\displaystyle +\)        
 \(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)  

 \(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)

 

2. Второе действие.

Умножаем десятки числа \(\displaystyle 5{\underline 4}8\) (это число \(\displaystyle 4\)) на число \(\displaystyle 45974{\small:}\)

 

\(\displaystyle 45974 \cdot 4=183896{\small.}\)

 

Записываем результат так, чтобы единицы числа \(\displaystyle 18389{\underline 6}\) шли под той цифрой, на которую мы умножали (в нашем случае это десятки числа \(\displaystyle 5{\underline 4}8{\small,}\) то есть число \(\displaystyle 4\)).

 

    \(\displaystyle 4\)\(\displaystyle 5\)\(\displaystyle 9\)\(\displaystyle 7\)\(\displaystyle 4\)
   x     
      \(\displaystyle 5\)\(\displaystyle \bf4\)\(\displaystyle 8\)

   \(\displaystyle 3\)\(\displaystyle 6\)\(\displaystyle 7\)\(\displaystyle 7\)\(\displaystyle 9\)\(\displaystyle 2\)
\(\displaystyle +\)        
  \(\displaystyle \bf1\)\(\displaystyle \bf8\)\(\displaystyle \bf3\)\(\displaystyle \bf8\)\(\displaystyle \bf9\)\(\displaystyle \bf6\) 
\(\displaystyle +\)        
 \(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)  

 \(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)

 

3. Третье действие.

Умножаем сотни числа \(\displaystyle {\underline 5}48\) (это число \(\displaystyle 5\)) на число \(\displaystyle 45974{\small:}\)

 

\(\displaystyle 45974 \cdot 5=229870{\small.}\)

 

Записываем результат так, чтобы единицы \(\displaystyle 22987{\underline 0}\) шли под той цифрой, на которую мы умножали (в нашем случае это сотни числа \(\displaystyle {\underline 5}48{\small,}\) то есть число \(\displaystyle 5\)).

 

    \(\displaystyle 4\)\(\displaystyle 5\)\(\displaystyle 9\)\(\displaystyle 7\)\(\displaystyle 4\)
   x     
      \(\displaystyle \bf5\)\(\displaystyle 4\)\(\displaystyle 8\)

   \(\displaystyle 3\)\(\displaystyle 6\)\(\displaystyle 7\)\(\displaystyle 7\)\(\displaystyle 9\)\(\displaystyle 2\)
\(\displaystyle +\)        
  \(\displaystyle 1\)\(\displaystyle 8\)\(\displaystyle 3\)\(\displaystyle 8\)\(\displaystyle 9\)\(\displaystyle 6\) 
\(\displaystyle +\)        
 \(\displaystyle \bf2\)\(\displaystyle \bf2\)\(\displaystyle \bf9\)\(\displaystyle \bf8\)\(\displaystyle \bf7\)\(\displaystyle \bf0\)  

 \(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)

 

4. Четвертое действие.

Складываем получившиеся числа:

 

   \(\displaystyle 3\)\(\displaystyle 6\)\(\displaystyle 7\)\(\displaystyle 7\)\(\displaystyle 9\)\(\displaystyle 2\)
\(\displaystyle +\)        
  \(\displaystyle 1\)\(\displaystyle 8\)\(\displaystyle 3\)\(\displaystyle 8\)\(\displaystyle 9\)\(\displaystyle 6\) 
\(\displaystyle +\)        
 \(\displaystyle 2\)\(\displaystyle 2\)\(\displaystyle 9\)\(\displaystyle 8\)\(\displaystyle 7\)\(\displaystyle 0\)  

 \(\displaystyle \bf2\)\(\displaystyle \bf5\)\(\displaystyle \bf1\)\(\displaystyle \bf9\)\(\displaystyle \bf3\)\(\displaystyle \bf7\)\(\displaystyle \bf5\)\(\displaystyle \bf2\)

 

 

В итоге имеем:

    \(\displaystyle 4\)\(\displaystyle 5\)\(\displaystyle 9\)\(\displaystyle 7\)\(\displaystyle 4\)
   x     
      \(\displaystyle 5\)\(\displaystyle 4\)\(\displaystyle 8\)

   \(\displaystyle 3\)\(\displaystyle 6\)\(\displaystyle 7\)\(\displaystyle 7\)\(\displaystyle 9\)\(\displaystyle 2\)
\(\displaystyle +\)        
  \(\displaystyle 1\)\(\displaystyle 8\)\(\displaystyle 3\)\(\displaystyle 8\)\(\displaystyle 9\)\(\displaystyle 6\) 
\(\displaystyle +\)        
 \(\displaystyle 2\)\(\displaystyle 2\)\(\displaystyle 9\)\(\displaystyle 8\)\(\displaystyle 7\)\(\displaystyle 0\)  

 \(\displaystyle \bf2\)\(\displaystyle \bf5\)\(\displaystyle \bf1\)\(\displaystyle \bf9\)\(\displaystyle \bf3\)\(\displaystyle \bf7\)\(\displaystyle \bf5\)\(\displaystyle \bf2\)

 

Ответ: \(\displaystyle 25193752{\small.}\)