To determine the least number of packages of caps and shirts that need to be ordered for the science club, we need to calculate the number of packages required for both items separately and then find the least common multiple (LCM) of those numbers.
The number of packages of caps needed can be found by dividing the total number of science club members (45) by the number of caps in each package (3):
45 ÷ 3 = 15.
So, 15 packages of caps will be needed.
Similarly, the number of packages of shirts needed can be found by dividing the total number of science club members (45) by the number of shirts in each package (5):
45 ÷ 5 = 9.
So, 9 packages of shirts will be needed.
To find the least number of packages that will need to be ordered, we need to find the LCM of 15 and 9. The LCM is the smallest number that is evenly divisible by both 15 and 9.
The prime factorization of 15 is 3 * 5, and the prime factorization of 9 is 3 * 3. The LCM can be found by taking the highest power of each prime factor:
LCM = 3 * 3 * 5 = 45.
Therefore, the least number of packages of caps and shirts that will need to be ordered is 45.