Problem Overview
Dr. Johnson is in need of purchasing an equal quantity of needles and syringes. The syringes are available in packs of 6, while the needles come in packs of 10.
Objective
The goal is to determine the smallest number of packages of syringes and needles Dr. Johnson can buy to ensure he has an identical number of each item.
Solution
To find the solution, we need to identify the smallest number that is a multiple of both 6 and 10.
Finding the Least Common Multiple
The factors of 6 are 2 and 3 (6 = 2 × 3), while the factors of 10 are 2 and 5 (10 = 2 × 5). The least common multiple (LCM) can be calculated by multiplying the highest powers of all prime factors involved.
The LCM of 6 and 10 is calculated as follows:
2 (from both) × 3 (from 6) × 5 (from 10) = 30.
Calculating Package Requirements
To determine the number of packages Dr. Johnson needs:
– For syringes: 30 (total required) ÷ 6 (syringes per package) = 5 packages.
– For needles: 30 (total required) ÷ 10 (needles per package) = 3 packages.
Conclusion
In summary, Dr. Johnson should purchase 5 packages of syringes and 3 packages of needles to achieve an equal quantity of both supplies.