Final answer:
The question that involves finding a common multiple of 12 and 20 is related to determining how many packages of pens and pencils are needed to have an equal number of each. The least common multiple (LCM) of 12 and 20 is found to be 60, which is used to form gift sets with equal numbers of pens and pencils.
Step-by-step explanation:
The question asking which scenario involves finding a common multiple of 12 and 20 is: 'How many packages of pens and how many packages of pencils are needed to have the same number of pens as pencils?' This involves finding a number that both 12 and 20 can divide into without leaving a remainder, which is known as a common multiple. The least common multiple (LCM) of 12 and 20 would be the smallest number they both can divide into and it is relevant when trying to make gift sets with the same amount of pens and pencils in each, using all available pens and pencils.
To find the LCM of 12 and 20, we could list the multiples of each number until we find a match, or use the prime factorization method:
- Prime factors of 12 are 22 × 3.
- Prime factors of 20 are 22 × 5.
- Take the highest powers of all prime factors: 22 (from both 12 and 20) and then 3 and 5 from 12 and 20 respectively.
- Multiply these together: 22 × 3 × 5 = 4 × 3 × 5 = 60.
So, the LCM of 12 and 20 is 60, which is the number of pens and pencils needed to form gift sets with an equal number of each, using all the pens and pencils.