Finding N Value In M = 2^n * 5^2 * 7 With 30 Divisors

Hey guys! Let's dive into a cool math problem today. We're going to explore how to find the value of 'n' in a number 'm' that's expressed in a special way. This involves understanding the relationship between a number's prime factors and its total number of divisors. Buckle up, because we're about to unravel this mathematical mystery!

Understanding the Problem

The problem states that we have a number m which is equal to 2 raised to the power of n, multiplied by 5 squared, and then multiplied by 7. In mathematical terms, this looks like: m = 2^n * 5^2 * 7. We are also given a crucial piece of information: the total number of divisors of m is 30. Our mission, should we choose to accept it, is to determine the value of n. This type of problem beautifully combines number theory concepts, especially prime factorization and divisor counting, making it a fantastic exercise for our mathematical minds.

Prime Factorization: The Key to Divisors

At the heart of this problem lies the concept of prime factorization. Every integer greater than 1 can be uniquely expressed as a product of prime numbers raised to certain powers. Prime numbers, like 2, 3, 5, and 7, are the building blocks of all other numbers. When we break down a number into its prime factors, we gain valuable insight into its divisors. In our case, m is already given in its prime factorized form: 2^n * 5^2 * 7. This means we know exactly which prime numbers divide m (2, 5, and 7) and the powers to which they appear.

The Divisor Counting Formula

Now, here’s where the magic happens. There’s a neat formula that allows us to calculate the total number of divisors of a number, based on its prime factorization. If a number x can be expressed as p1^a * p2^b * p3^c… where p1, p2, p3… are distinct prime numbers and a, b, c… are their respective exponents, then the total number of divisors of x is given by: (a + 1) * (b + 1) * (c + 1)…

This formula works because any divisor of x will be formed by taking each prime factor raised to a power between 0 and its exponent in the prime factorization of x. For example, if we have p1^a, we can choose p1^0, p1^1, p1^2, …, up to p1^a, giving us a + 1 choices. We do this for each prime factor and multiply the number of choices together to get the total number of divisors. It’s a pretty slick trick, right?

Applying the Formula to Our Problem

Let’s apply this formula to our number m. We know that m = 2^n * 5^2 * 7, which can also be written as 2^n * 5^2 * 7^1. So, the exponents of our prime factors are n, 2, and 1. According to the divisor counting formula, the total number of divisors of m is (n + 1) * (2 + 1) * (1 + 1). We are told that this number is equal to 30. Therefore, we have the equation:

(n + 1) * (2 + 1) * (1 + 1) = 30

Solving the Equation

Now that we have our equation, it’s time to put our algebra skills to work and solve for n. The equation we have is:

(n + 1) * (2 + 1) * (1 + 1) = 30

Let’s simplify this step by step.

Simplifying the Equation

First, let's simplify the constants inside the parentheses:

(n + 1) * 3 * 2 = 30

Now we can multiply the 3 and the 2 together:

(n + 1) * 6 = 30

Isolating (n + 1)

Our next goal is to isolate the term (n + 1). To do this, we can divide both sides of the equation by 6:

(n + 1) * 6 / 6 = 30 / 6

n + 1 = 5

Solving for n

We're almost there! To find the value of n, we simply subtract 1 from both sides of the equation:

n + 1 - 1 = 5 - 1

n = 4

So, we've found that the value of n is 4. Awesome!

Verifying the Solution

To make sure we haven't made any mistakes, let's plug our value of n back into the original equation and see if it holds true. We found that n = 4, so let's substitute that into the expression for m:

m = 2^4 * 5^2 * 7

m = 16 * 25 * 7

m = 2800

Now, let's calculate the number of divisors of 2800 using our formula. The prime factorization of 2800 is indeed 2^4 * 5^2 * 7^1. So, the number of divisors should be:

(4 + 1) * (2 + 1) * (1 + 1) = 5 * 3 * 2 = 30

This matches the information given in the problem! We have successfully verified that our solution, n = 4, is correct. High five!

Conclusion: The Power of Prime Factorization

In this mathematical adventure, we've successfully found the value of n in the equation m = 2^n * 5^2 * 7, given that m has 30 divisors. By understanding the power of prime factorization and the divisor counting formula, we were able to break down the problem into manageable steps and arrive at the solution n = 4. This problem highlights how fundamental concepts in number theory can be used to solve interesting and challenging questions.

So, the next time you encounter a problem involving divisors, remember the power of prime factorization and that handy divisor counting formula. You'll be able to tackle it like a pro! Keep exploring, keep learning, and most importantly, keep having fun with math! You guys rock!