A jug holds 7 liters of water and a mug holds 3 liters. A bottle contains 34 liters. Find all possible combinations of jugs and mugs that can fill the bottle exactly. Also, state the combination that uses the least total number of containers.

Step-by-step solution

1. Form the equation $7j+3m=34$ where $j$ and $m$ are non‑negative integers.
2. Since $j\le \lfloor 34/7\rfloor =4$, test $j=0$ to $4$.
3. $j=0$ gives $3m=34$, no integer solution.
4. $j=1$ gives $7+3m=34\Rightarrow 3m=27\Rightarrow m=9$.
5. $j=2$ gives $14+3m=34\Rightarrow 3m=20$, no integer.
6. $j=3$ gives $21+3m=34\Rightarrow 3m=13$, no integer.
7. $j=4$ gives $28+3m=34\Rightarrow 3m=6\Rightarrow m=2$.
8. Thus two valid combinations: $(j,m)=(1,9)$ and $(4,2)$.
9. Count containers: first uses $10$ ($1+9$), second uses $6$ ($4+2$).
10. The least total number of containers is $6$, achieved with $4$ jugs and $2$ mugs.

Concept tested: finding all integer solutions to a linear equation

Published 7 October 2026 · How this solution was prepared and checked