Ratio and Divisibility Constraints

Let weekly coins collected by A and B be $3k$ and $4k$ respectively for some positive integer $k$.

1. Coins collected by A in 5 weeks = $5 \times 3k = 15k$. $15k$ is a multiple of $7 \implies k$ is a multiple of $7$.

2. Coins collected by B in 3 weeks = $3 \times 4k = 12k$. $12k$ is a multiple of $24 \implies k$ is an even number.

Thus, $k$ must be a multiple of both 7 and 2 $\implies k$ is a multiple of $14$.

The minimum possible positive integer $k$ is $14$.

Minimum coins collected by A in one week = $3k = 3 \times 14 = 42$.