To find the greatest number of identical packages Marcus can make using all of the 36 bars of soap and 24 toothbrushes, we need to determine the greatest common divisor (GCD) of the two numbers.
-
Find the factors of each number:
- For 36: The factors are 1, 2, 3, 4, 6, 9, 12, 18, 36.
- For 24: The factors are 1, 2, 3, 4, 6, 8, 12, 24.
-
Identify the common factors:
- The common factors of 36 and 24 are 1, 2, 3, 4, 6, and 12.
-
Determine the greatest common factor:
- The greatest common factor is 12.
Thus, the greatest number of identical packages that Marcus can make using all of the soap and toothbrushes is 12 packages.
In each package, he would include:
- \( \frac{36 \text{ bars of soap}}{12 \text{ packages}} = 3 \text{ bars of soap per package} \)
- \( \frac{24 \text{ toothbrushes}}{12 \text{ packages}} = 2 \text{ toothbrushes per package} \)
So, each of the 12 packages would contain 3 bars of soap and 2 toothbrushes.