1. **State the problem:** We need to find the smallest of four consecutive multiples of 7 whose sum is 1582. 2. **Define variables:** Let the smallest multiple be $7n$, where $n$ is an integer. 3. **Express the next three multiples:** The four consecutive multiples are $7n$, $7(n+1)$, $7(n+2)$, and $7(n+3)$. 4. **Set up the equation for their sum:** $$7n + 7(n+1) + 7(n+2) + 7(n+3) = 1582$$ 5. **Simplify the equation:** $$7n + 7n + 7 + 7n + 14 + 7n + 21 = 1582$$ $$28n + (7 + 14 + 21) = 1582$$ $$28n + 42 = 1582$$ 6. **Solve for $n$:** $$28n = 1582 - 42$$ $$28n = 1540$$ $$n = \frac{1540}{28}$$ $$n = 55$$ 7. **Find the smallest multiple:** $$7n = 7 \times 55 = 385$$ **Answer:** The smallest of the four consecutive multiples of 7 is **385**.