Divisibility Problem: Find The Product P

Let's dive into a fun little math problem that involves divisibility and natural numbers. If you're scratching your head already, don't worry, we'll break it down piece by piece. The problem states: if p is the product of all natural numbers n for which (n + 1) divides 6, then what is the value of p? Sounds like a mouthful, right? Well, let's simplify it and figure out how to approach it like a pro.

Understanding the Question

First off, what are we even looking for? We need to find all the natural numbers n such that when you add 1 to them, the result divides 6. Remember, a natural number is a positive whole number (1, 2, 3, and so on). So, we're searching for n values where (n + 1) goes evenly into 6. Once we find all those n values, we multiply them together, and that product is what we call p. This question tests your understanding of divisibility, natural numbers, and basic algebraic thinking. Now that we're clear on what the question is asking, we can dive into the steps to solve it.

Finding Possible Values of n

Okay, so (n + 1) has to divide 6. That means (n + 1) can only be certain numbers. What are the divisors of 6? Well, 6 can be divided evenly by 1, 2, 3, and 6. So, (n + 1) could be any of these numbers. Let's explore each possibility:

1. If (n + 1) = 1, then n = 0. But hold on! Remember that n has to be a natural number. Natural numbers start at 1, not 0. So, n = 0 doesn't fit the bill.
2. If (n + 1) = 2, then n = 1. This is a natural number, so n = 1 is a valid solution.
3. If (n + 1) = 3, then n = 2. Another natural number! So, n = 2 is also a valid solution.
4. If (n + 1) = 6, then n = 5. This is indeed a natural number, making n = 5 a valid solution.

So, the natural numbers n that satisfy the condition are 1, 2, and 5. Seems pretty straightforward when you break it down like that, right?

Calculating the Product p

Now that we've found all the possible values of n, it's time to calculate p. Remember, p is the product of all these n values. So, we just need to multiply them together:

p = 1 * 2 * 5 = 10

Therefore, the value of p is 10. That's it! We've found the product of all natural numbers n that satisfy the given condition. This type of problem is common in math competitions and helps reinforce fundamental concepts. Keep practicing!

Let's Do Some More Examples!

To really nail this concept, let's explore some more examples that are similar, but with a few twists. This way, you'll be better prepared to tackle similar problems in the future. This skill is very helpful for test preparation.

Example 1: Divisibility by 12

Suppose we change the number to 12. The question becomes: if p is the product of the natural numbers n for which (n + 1) divides 12, then what is the value of p?

First, we need to find the divisors of 12. These are 1, 2, 3, 4, 6, and 12. Now, we check which of these can be equal to (n + 1):

• If (n + 1) = 1, then n = 0 (not a natural number).
• If (n + 1) = 2, then n = 1 (natural number).
• If (n + 1) = 3, then n = 2 (natural number).
• If (n + 1) = 4, then n = 3 (natural number).
• If (n + 1) = 6, then n = 5 (natural number).
• If (n + 1) = 12, then n = 11 (natural number).

So, the natural numbers n are 1, 2, 3, 5, and 11. Now, we calculate p:

p = 1 * 2 * 3 * 5 * 11 = 330

Therefore, in this case, the value of p is 330.

Example 2: Divisibility by 15

Let's try another one. If p is the product of the natural numbers n for which (n + 1) divides 15, then what is the value of p?

The divisors of 15 are 1, 3, 5, and 15. Let's find the corresponding values of n:

• If (n + 1) = 1, then n = 0 (not a natural number).
• If (n + 1) = 3, then n = 2 (natural number).
• If (n + 1) = 5, then n = 4 (natural number).
• If (n + 1) = 15, then n = 14 (natural number).

So, the natural numbers n are 2, 4, and 14. Now, we calculate p:

p = 2 * 4 * 14 = 112

In this example, the value of p is 112.

Key Takeaways

• Divisibility: Understanding what it means for one number to divide another is crucial. Remember that a number a divides a number b if b can be divided by a without any remainder.
• Natural Numbers: Always remember the definition of natural numbers. They are positive whole numbers starting from 1. Zero is not a natural number.
• Systematic Approach: Break down the problem into smaller, manageable steps. First, identify the divisors of the given number. Then, find the corresponding values of n. Finally, calculate the product p.
• Practice: The more you practice, the better you'll become at solving these types of problems. Try different numbers and see how the process works.

Wrapping Up

So, there you have it! We've tackled a divisibility problem, found the product of natural numbers that satisfy a given condition, and even explored some extra examples. Remember, math isn't about memorizing formulas, it's about understanding the underlying concepts and applying them in a logical way. Keep practicing, keep exploring, and you'll be a math whiz in no time! Feel free to try out these problems with different numbers, math is all about discovery, after all!

These types of questions not only help you with math competitions, but they also sharpen your problem-solving skills, which are valuable in any field. Keep up the great work!