Let $$n$$ be the least positive integer such that $$168$$ is a factor of $$1134^n$$. If $$m$$ is the least positive integer such that $$1134^n$$ is a factor of $$168^m$$, then $$m+n$$ equals:

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Accepted Answer

The prime factorizations are as follows:  
- $$168 = 2^3 \cdot 3 \cdot 7$$  
- $$1134 = 2 \cdot 3^4 \cdot 7$$

1. **Find the smallest positive value of $$n$$**:  
   For $$168$$ to divide $$1134^n$$, the powers of each prime factor in $$1134^n$$ must be greater than or equal to those in $$168$$:  
   - For prime factor $$2$$: $$ n \geq 3 $$ (since $$1 \cdot n \geq 3$$)  
   - For prime factor $$3$$: $$ n \geq 1 $$ (since $$4 \cdot n \geq 1$$)  
   - For prime factor $$7$$: $$ n \geq 1 $$.

Thus, the smallest value of $$ n = 3 $$.

2. **Find the smallest positive value of $$m$$**:  
   For $$1134^n$$ to divide $$168^m$$, the powers of each prime factor in $$168^m$$ must be greater than or equal to those in $$1134^n$$:  
   - For prime factor $$2$$: $$ m \geq n = 12 $$.

3. **Calculate $$m+n$$**:  
   $$
   m+n = 12+3 = 15.
   $$

Thus, the answer is **15**.

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