A ribbon can be cut exactly into pieces of 9 cm or exactly into pieces of 12 cm. What is the shortest possible length of the ribbon?

Step-by-step solution

1. 1

   Translate the words into a divisibility statement. "Cuts exactly into 9 cm pieces" means the length $L$ is a multiple of $9$; "exactly into 12 cm pieces" means $L$ is a multiple of $12$. So $L$ is a common multiple of $9$ and $12$, and "shortest" makes it the least common multiple - not the greatest common divisor, which is the classic mix-up here.

2. 2

   Factor both numbers into primes.

   $9=3^2,\qquad 12=2^2\times 3$

3. 3

   Take the highest power of each prime. For an LCM you keep the largest exponent of every prime that appears in either number: $2^2$ from $12$, and $3^2$ from $9$.

   $\operatorname{lcm}(9,12)=2^2\times 3^2=4\times 9=36$

4. 4

   Check both divisions come out whole.

   $36\div 9=4\ \text{pieces},\qquad 36\div 12=3\ \text{pieces}$

   Both are whole numbers, so $36$ cm genuinely works.

5. 5

   Confirm nothing smaller works. The multiples of $12$ below $36$ are $12$ and $24$; neither is divisible by $9$ ($12\div9$ and $24\div9$ are not whole). So $36$ cm is the shortest ribbon ✓.