A number is divided successively by 3, 4, and 7, leaving remainders 2, 1, and 4 respectively. Determine the remainder when the same number is divided by 84.

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To solve this problem:

1. Represent the number as $$ x $$. Using the Chinese Remainder Theorem:  
   - From the first division by 3, $$ x = 3k + 2 $$.  
   - From the second division by 4, $$ x = 4m + 1 $$.  
   - From the third division by 7, $$ x = 7n + 4 $$.

2. Solve these equations step-by-step:  
   - Start with $$ z = 7 	imes 1 + 4 = 11 $$ (from the third division).  
   - Substitute $$ z $$ into the second equation: $$ y = 4 	imes z + 1 = 4 	imes 11 + 1 = 45 $$.  
   - Substitute $$ y $$ into the first equation: $$ x = 3 	imes y + 2 = 3 	imes 45 + 2 = 137 $$.

3. Finally, divide $$ x = 137 $$ by $$ 84 $$:  
   - Perform the division: $$ x = 137 \div 84 $$, which gives a remainder of **53**.

Thus, when the number is divided by $$ 84 $$, the remainder is **53**.

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