We are given that the DECI-Pizza Company has a pizza divided into 10 slices with the following proportions:

• Two slices are each $\frac{1}{6}$ of the pizza.
• Two slices are each $\frac{1}{8}$.
• Four slices are each $\frac{1}{12}$.
• Two slices are each $\frac{1}{24}$.

The total pizza must be shared equally among $n$ friends, and we need to determine for which values of $n > 1$ this is possible.

Step 1: Calculate the total fraction of the pizza

Let's first calculate the total amount of pizza, which should sum up to 1.

• Two slices of $\frac{1}{6}$ contribute: $2 \times \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$
• Two slices of $\frac{1}{8}$ contribute: $2 \times \frac{1}{8} = \frac{2}{8} = \frac{1}{4}$
• Four slices of $\frac{1}{12}$ contribute: $4 \times \frac{1}{12} = \frac{4}{12} = \frac{1}{3}$
• Two slices of $\frac{1}{24}$ contribute: $2 \times \frac{1}{24} = \frac{2}{24} = \frac{1}{12}$

Now, let's sum all the contributions: $\frac{1}{3} + \frac{1}{4} + \frac{1}{3} + \frac{1}{12}$

To add these fractions, we need a common denominator. The least common denominator of 3, 4, and 12 is 12.

• $\frac{1}{3} = \frac{4}{12}$
• $\frac{1}{4} = \frac{3}{12}$
• $\frac{1}{3} = \frac{4}{12}$
• $\frac{1}{12} = \frac{1}{12}$

Summing these: $\frac{4}{12} + \frac{3}{12} + \frac{4}{12} + \frac{1}{12} = \frac{12}{12} = 1$ Thus, the total pizza is indeed the whole pizza.

Step 2: Determine the values of $n$

Now, the task is to distribute the pizza equally among $n$ friends. Each friend should receive an equal fraction of the total pizza.

The slices are $\frac{1}{6}$, $\frac{1}{8}$, $\frac{1}{12}$, and $\frac{1}{24}$, and we cannot cut the slices. The total pizza must be divided equally, so the total fraction each person receives must be a divisor of 1.

Let's check the smallest slice size, which is $\frac{1}{24}$. This indicates that $n$ must divide the total number of "units" in the pizza, where each "unit" corresponds to $\frac{1}{24}$ of the pizza.

Thus, $n$ must divide the least common multiple (LCM) of the denominators: 6, 8, 12, and 24.

Step 3: Find the least common multiple (LCM)

The LCM of 6, 8, 12, and 24 is calculated by finding the prime factorizations:

• $6 = 2 \times 3$
• $8 = 2^3$
• $12 = 2^2 \times 3$
• $24 = 2^3 \times 3$

The LCM is obtained by taking the highest powers of all prime factors: $\text{LCM}(6, 8, 12, 24) = 2^3 \times 3 = 24$

Thus, the total pizza can be divided into 24 equal parts.

Step 4: Determine the divisors of 24

The values of $n$ that allow the pizza to be shared equally are the divisors of 24. These divisors are: $1, 2, 3, 4, 6, 8, 12, 24$

Since $n > 1$, the possible values of $n$ are: $n = 2, 3, 4, 6, 8, 12, 24$

Conclusion:

The possible values of $n > 1$ for which the pizza can be shared equally among $n$ friends are: $n = 2, 3, 4, 6, 8, 12, 24$

Would you like more details or have any questions? Here are five related questions:

1. How do you calculate the least common multiple (LCM) for other sets of numbers?
2. Why can't slices be further divided in this problem?
3. How would the solution change if different slice sizes were introduced?
4. What if the total number of slices was different, but with the same proportions?
5. How would the solution be affected if $n = 5$ friends shared the pizza?

Tip: Always check the LCM when dividing portions equally across multiple people without cutting pieces further.