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Теория: Деление натуральных чисел на трехзначные числа

Задание

По известному процессу деления в столбик найти частное (то есть найти пропущенные цифры):

 \(\displaystyle 7\)\(\displaystyle 3\)\(\displaystyle 0\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 2\)\(\displaystyle 3\)\(\displaystyle 8\)
\(\displaystyle –\)        
 \(\displaystyle 7\)\(\displaystyle 1\)\(\displaystyle 4\)  
  \(\displaystyle 1\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 6\)   
 \(\displaystyle –\)       
  \(\displaystyle 1\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 6\)   
     \(\displaystyle 0\)   

 

Решение

Найдем пропущенные цифры при делении в столбик:

 \(\displaystyle 7\)\(\displaystyle 3\)\(\displaystyle 0\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 2\)\(\displaystyle 3\)\(\displaystyle 8\)
\(\displaystyle –\)        
 \(\displaystyle 7\)\(\displaystyle 1\)\(\displaystyle 4\)  \(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)
  \(\displaystyle 1\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 6\)   
 \(\displaystyle –\)       
  \(\displaystyle 1\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 6\)   
     \(\displaystyle 0\)   

 

1. Под \(\displaystyle 730\) стоит число \(\displaystyle 714={\bf 3}\cdot 238{\small.}\) Следовательно, первая неизвестная цифра частного – это \(\displaystyle 3{\small:}\)

 

 \(\displaystyle 7\)\(\displaystyle 3\)\(\displaystyle 0\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 2\)\(\displaystyle 3\)\(\displaystyle 8\)
\(\displaystyle –\)        
 \(\displaystyle \bf 7\)\(\displaystyle \bf 1\)\(\displaystyle \bf 4\)  \(\displaystyle \bf 3\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)
  \(\displaystyle 1\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 6\)   
 \(\displaystyle –\)       
  \(\displaystyle 1\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 6\)   
     \(\displaystyle 0\)   

 

2. Далее вычитаем из \(\displaystyle 730\) число \(\displaystyle 714\) (\(\displaystyle 730-714=16\)) и сносим цифру из разряда десятков числа \(\displaystyle 730{\bf6}6\) (это цифра \(\displaystyle 6\)). Получаем число \(\displaystyle 166{\small.}\)

 

 \(\displaystyle 7\)\(\displaystyle 3\)\(\displaystyle 0\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 2\)\(\displaystyle 3\)\(\displaystyle 8\)
\(\displaystyle –\)        
 \(\displaystyle 7\)\(\displaystyle 1\)\(\displaystyle 4\)  \(\displaystyle 3\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)
  \(\displaystyle \bf 1\)\(\displaystyle \bf 6\)\(\displaystyle \bf 6\)\(\displaystyle 6\)   
 \(\displaystyle –\)       
  \(\displaystyle 1\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 6\)   
     \(\displaystyle 0\) 

Число \(\displaystyle 166\) меньше \(\displaystyle 238{\small,}\) поэтому сносим следующий разряд (то есть единицы числа \(\displaystyle 7306{\bf6}{\small,}\)) и  записываем \(\displaystyle {\bf 0}\) в частном. Поэтому вторая неизвестная цифра частного – это \(\displaystyle 0{\small.}\)

 

 \(\displaystyle 7\)\(\displaystyle 3\)\(\displaystyle 0\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 2\)\(\displaystyle 3\)\(\displaystyle 8\)
\(\displaystyle –\)        
 \(\displaystyle 7\)\(\displaystyle 1\)\(\displaystyle 4\)  \(\displaystyle 3\)\(\displaystyle 0\)\(\displaystyle {\small?}\)
  \(\displaystyle 1\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 6\)   
 \(\displaystyle –\)       
  \(\displaystyle 1\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 6\)   
     \(\displaystyle 0\) 

 

3. Под числом \(\displaystyle 1666\) стоит число \(\displaystyle 1666{\small.}\) Чтобы получить число \(\displaystyle 1666{\small,}\) надо \(\displaystyle 238\) умножить на \(\displaystyle 7\) (\(\displaystyle 1666=238\cdot {\bf 7} \)). Следовательно, третья неизвестная цифра частного – это \(\displaystyle 7{\small.}\)

 

 \(\displaystyle 7\)\(\displaystyle 3\)\(\displaystyle 0\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 2\)\(\displaystyle 3\)\(\displaystyle 8\)
\(\displaystyle –\)        
 \(\displaystyle 7\)\(\displaystyle 1\)\(\displaystyle 4\)  \(\displaystyle 3\)\(\displaystyle 0\)\(\displaystyle \bf 7\)
  \(\displaystyle 1\)\(\displaystyle 6\)\(\displaystyle 6\)\(\displaystyle 6\)   
 \(\displaystyle –\)       
  \(\displaystyle \bf 1\)\(\displaystyle \bf 6\)\(\displaystyle \bf 6\)\(\displaystyle \bf 6\)   
     \(\displaystyle 0\) 

Ответ: \(\displaystyle 307{\small.}\)