Alright, let's break down this problem step-by-step so you can totally understand how to solve it. This looks like a fun little puzzle involving some basic algebra and pattern recognition. So, if abc = 6 and cde = 60, we need to figure out what def equals. Let's get started!

Understanding the Problem

First off, it’s essential to realize what these expressions mean. Here, abc, cde, and def aren't typical algebraic variables. Instead, they represent the product of individual variables a, b, c, d, e, and f. So, we can rewrite the given equations as:

• a * b * c = 6
• c * d * e = 60

Our mission is to find the value of:

• d * e * f = ?

To solve this, we need to find a relationship between the known equations and the unknown one. Let's dive deeper into analyzing these equations.

Breaking Down the Given Equations

Let's think about the factors of the numbers we have. The number 6 can be factored into several combinations, such as 1 * 1 * 6, 1 * 2 * 3, or 2 * 1 * 3. Similarly, 60 can be factored into combinations like 1 * 1 * 60, 3 * 4 * 5, or 2 * 5 * 6. The key here is to find a common factor between the two given equations because 'c' appears in both.

Identifying Common Factors

Since c is present in both equations, we need to find a factorization of 6 and 60 that allows c to have a consistent value. Let's list out some possibilities:

• Factors of 6: 1, 2, 3, 6
• Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

From these, we can see that 1, 2, 3, and 6 are common factors. This gives us a starting point to test different values for c.

Solving for def

Okay, let's walk through a methodical approach to crack this problem. We'll explore possible values for c and see how they influence the values of the other variables.

Case 1: Assume c = 1

If we assume c = 1, then from the first equation a * b * c = 6, we get a * b * 1 = 6, which means a * b = 6. From the second equation c * d * e = 60, we get 1 * d * e = 60, so d * e = 60. This doesn't immediately lead us to a clear value for def, but let's keep it in mind.

Case 2: Assume c = 2

If c = 2, then a * b * 2 = 6, so a * b = 3. Also, 2 * d * e = 60, which means d * e = 30. Still not quite there, but we're narrowing it down.

Case 3: Assume c = 3

Now, let's try c = 3. Then a * b * 3 = 6, so a * b = 2. And 3 * d * e = 60, which gives us d * e = 20. Getting closer!

Case 4: Assume c = 6

Finally, let's try c = 6. Then a * b * 6 = 6, so a * b = 1. And 6 * d * e = 60, which means d * e = 10. This looks promising.

Finding the Pattern

Notice that when c = 6, we have d * e = 10. We need one more piece of information to find def. Let's express def as d * e * f. Since we know d * e = 10, we can write def = 10 * f.

To find f, we need to look for a pattern or relationship between the equations. We have:

• a * b * c = 6
• c * d * e = 60
• d * e * f = ?

Let's consider the ratios between these expressions:

• (c * d * e) / (a * b * c) = 60 / 6 = 10
• This simplifies to (d * e) / (a * b) = 10

Now, if we assume a * b = 1 (as we found when c = 6), then d * e = 10, which we already know.

So, the question is: How does f relate to the other variables? To find this, we need to consider the context or any hidden assumptions in the problem.

Making an Educated Guess

Without additional information, let's make a logical leap based on the alphabetical order of the variables. Since the problem presents a sequence abc, cde, and def, it's plausible that there's a simple multiplicative relationship.

We know abc = 6 and cde = 60. The jump from 6 to 60 is a factor of 10. If we assume the same multiplicative relationship holds, then the jump from cde = 60 to def should also be a factor. However, this is a bit too simplistic and may not be correct.

Another Approach: Prime Factorization

Let’s use prime factorization to analyze the numbers:

• 6 = 2 * 3
• 60 = 2^2 * 3 * 5

If abc = 2 * 3 and cde = 2^2 * 3 * 5, we can try assigning values:

• Let a = 1, b = 1, c = 6. Then abc = 1 * 1 * 6 = 6.
• Since c = 6, cde = 60, so 6 * d * e = 60, thus d * e = 10. Let d = 2, e = 5. Then cde = 6 * 2 * 5 = 60.

Now we need to find f such that def follows a logical pattern. If we continue the prime factorization pattern, we might expect def to be related to the next prime number or a combination of existing ones.

Attempting a Pattern Extension

If d = 2 and e = 5, then def = 2 * 5 * f. We need to find f. Let's observe the transition from abc to cde:

• abc = 1 * 1 * 6
• cde = 6 * 2 * 5

There isn't an immediately obvious pattern here, but let’s consider the possibility that f could be related to the next prime number after 5, which is 7. Then, def = 2 * 5 * 7 = 70.

However, without a clear and consistent pattern, this is still speculative.

Conclusion

Given the information, it’s challenging to find a definitive value for def. However, based on our analysis and the assumption that c = 6 (which gives d * e = 10), and making an educated guess based on the progression, one plausible answer could be:

If we assume the sequence involves multiplying by consecutive prime numbers or a similar logical progression, a reasonable guess for def is 70.

So, based on these assumptions: def = 70.

Final Answer: 70