The number of coins collected per week by two coin-collectors A and B are in the ratio 3:4. If the total number of coins collected by A in 5 weeks is a multiple of 7, and the total number of coins collected by B in 3 weeks is a multiple of 24, then the minimum possible number of coins collected by A in one week is:

Question

Accepted Answer

1. **Define Variables**:
   - Let the number of coins collected by A and B in a week be $$ 3x $$ and $$ 4x $$ respectively.

2. **Condition for A's Coins**:
   - Total coins collected by A in 5 weeks = $$ 15x $$.
   - For $$ 15x $$ to be a multiple of 7, $$ x $$ must be a multiple of 7.

3. **Condition for B's Coins**:
   - Total coins collected by B in 3 weeks = $$ 12x $$.
   - For $$ 12x $$ to be a multiple of 24, $$ x $$ must be a multiple of 2.

4. **Find the LCM**:
   - The least value of $$ x $$ that satisfies both conditions (multiple of 7 and multiple of 2) is:
     $$
     	ext{LCM}(7, 2) = 14
     $$

5. **Calculate Minimum Coins Collected by A**:
   - Minimum coins collected by A in one week = $$ 3x = 3 	imes 14 = 42 $$.

Thus, the minimum possible number of coins collected by A in one week is **42**.