Solving Φ * 3 * 5: A Step-by-Step Math Guide
Hey guys! Today, we're diving into a fun little math problem: φ * 3 * 5. This might look a bit intimidating at first, especially with that φ symbol, but don't worry, we'll break it down together. We're going to go through each step in detail, so you'll not only get the answer but also understand the process behind it. So, grab your calculators (or your mental math muscles) and let's get started!
Understanding the Basics
Before we jump into solving φ * 3 * 5, let's quickly recap some basic math concepts. This will make sure we're all on the same page and help you follow along more easily. We will define the Euler's totient function. We will review the order of operations (PEMDAS/BODMAS). And also we will mention the multiplication basics.
Euler's Totient Function (φ)
The φ symbol represents Euler's totient function, often denoted as φ(n). This function counts the number of positive integers less than or equal to n that are relatively prime to n. In simpler terms, it tells you how many numbers less than n don't share any common factors with n other than 1. For example, φ(8) = 4 because the numbers 1, 3, 5, and 7 are relatively prime to 8.
Calculating Euler's totient function can seem tricky, but there's a handy formula for it. If you know the prime factorization of n, say n = p₁^k₁ * p₂^k₂ * ... * pᵣ^kᵣ, then:
φ(n) = n * (1 - 1/p₁) * (1 - 1/p₂) * ... * (1 - 1/pᵣ)
Where p₁, p₂, ..., pᵣ are the distinct prime factors of n. This formula might look a bit complex, but it's quite straightforward once you get the hang of it. We will use this formula later when we need to calculate φ(15), which is crucial for solving our main problem.
Order of Operations (PEMDAS/BODMAS)
Remember PEMDAS/BODMAS? It's the golden rule for solving mathematical expressions! This acronym helps us remember the correct order in which to perform operations:
- Parentheses / Brackets
- Exponents / Orders
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
Following the order of operations ensures that we all arrive at the same correct answer. For our problem, φ * 3 * 5, we'll primarily be dealing with multiplication, but it's always good to keep PEMDAS/BODMAS in mind for more complex calculations.
Multiplication Basics
Multiplication is the backbone of our problem today. It's one of the basic arithmetic operations and represents repeated addition. For example, 3 * 5 means adding 3 to itself 5 times (or vice versa). We all know our times tables, right? 😜
In our expression, φ * 3 * 5, we'll be multiplying the value of the Euler's totient function by 3 and then by 5. A solid grasp of multiplication is key to making this process smooth and accurate.
Step-by-Step Solution for φ * 3 * 5
Now, let's get to the fun part – actually solving φ * 3 * 5! We'll break it down into manageable steps to make it super clear and easy to follow. The key here is to first figure out what φ represents in this context and then perform the multiplication. So, let's dive in!
Step 1: Understanding φ in the Expression
The first thing we need to clarify is what the argument of the Euler's totient function is. In the expression φ * 3 * 5, it's implied that we need to compute φ(15) because 3 * 5 = 15. So, we're actually trying to solve φ(15) * 3 * 5. Understanding this is crucial because it sets the stage for the rest of the calculation. Remember, φ(n) tells us how many numbers less than or equal to n are relatively prime to n.
Step 2: Calculating φ(15)
To calculate φ(15), we can use the formula we discussed earlier. First, we need to find the prime factorization of 15. The prime factors of 15 are 3 and 5, since 15 = 3 * 5. Now, we can apply the formula:
φ(15) = 15 * (1 - 1/3) * (1 - 1/5)
Let's break this down further:
- (1 - 1/3) = 2/3
- (1 - 1/5) = 4/5
So, φ(15) = 15 * (2/3) * (4/5). Now, we just need to multiply these fractions together. Multiplying fractions can be a breeze if you remember to multiply the numerators (top numbers) and the denominators (bottom numbers).
φ(15) = 15 * (2/3) * (4/5) = (15 * 2 * 4) / (3 * 5) = 120 / 15 = 8
Therefore, φ(15) = 8. This means there are 8 numbers less than or equal to 15 that are relatively prime to 15. These numbers are 1, 2, 4, 7, 8, 11, 13, and 14. Now that we've found φ(15), we're one step closer to solving our main problem!
Step 3: Substituting φ(15) into the Expression
Now that we know φ(15) = 8, we can substitute this value back into our original expression. Remember, we're trying to solve φ(15) * 3 * 5. So, we replace φ(15) with 8, giving us:
8 * 3 * 5
This looks much simpler, doesn't it? We've transformed a potentially confusing expression into a straightforward multiplication problem. This is a crucial step because it simplifies the calculation and makes it easier to manage. Substituting values is a common technique in math, and it's super helpful for breaking down complex problems.
Step 4: Performing the Multiplication
Finally, we get to the last step: performing the multiplication. We have 8 * 3 * 5. We can multiply these numbers in any order, thanks to the associative property of multiplication (which basically says that the way you group numbers when multiplying doesn't change the answer). Let's start by multiplying 8 and 3:
8 * 3 = 24
Now, we have:
24 * 5
Multiplying 24 by 5 is the final step. You can do this mentally, on paper, or with a calculator. If you're doing it mentally, you might think of it as (20 * 5) + (4 * 5) = 100 + 20 = 120. So:
24 * 5 = 120
And there we have it! We've successfully solved φ * 3 * 5. The final answer is 120.
Alternative Methods to Solve φ * 3 * 5
While we've walked through a step-by-step solution, there are often multiple paths to the same answer in math. Let's explore some alternative methods to solve φ * 3 * 5. This can help you deepen your understanding and give you more tools in your problem-solving toolkit. Knowing different methods can also make you a more flexible and confident mathematician. So, let's check out some cool alternatives!
Method 1: Direct Calculation of φ(15)
Instead of using the formula, we can directly count the numbers less than or equal to 15 that are relatively prime to 15. Remember, a number is relatively prime to 15 if it shares no common factors with 15 other than 1. The numbers we need to consider are 1, 2, 3, ..., 15. Let's eliminate the ones that share factors with 15 (which are multiples of 3 and 5):
- Multiples of 3: 3, 6, 9, 12, 15
- Multiples of 5: 5, 10
Now, let's list the numbers that are not multiples of 3 or 5:
1, 2, 4, 7, 8, 11, 13, 14
Counting these, we find there are 8 numbers. So, φ(15) = 8. This method is more hands-on and can be helpful for smaller numbers. It gives you a concrete sense of what the Euler's totient function is actually counting.
Method 2: Using the Property φ(mn) = φ(m)φ(n)
If m and n are relatively prime, then φ(mn) = φ(m)φ(n). Since 3 and 5 are prime numbers, they are relatively prime to each other. So, we can calculate φ(15) as follows:
φ(15) = φ(3 * 5) = φ(3) * φ(5)
Now, we need to find φ(3) and φ(5). For prime numbers, φ(p) = p - 1. So:
- φ(3) = 3 - 1 = 2
- φ(5) = 5 - 1 = 4
Therefore, φ(15) = 2 * 4 = 8. This method is particularly useful when dealing with the product of distinct prime numbers. It simplifies the calculation by breaking it down into smaller, more manageable parts.
Method 3: Rearranging the Multiplication
Once we have φ(15) = 8, instead of multiplying 8 * 3 first, we could multiply 3 * 5 first:
8 * 3 * 5 = 8 * (3 * 5) = 8 * 15
Now, we just need to multiply 8 by 15. You can break this down as (8 * 10) + (8 * 5) = 80 + 40 = 120. This method highlights that the order of multiplication doesn't affect the result, thanks to the commutative property of multiplication. Sometimes, rearranging the numbers can make the calculation easier, depending on your mental math preferences.
Real-World Applications of Euler's Totient Function
Okay, so we've nailed how to solve φ * 3 * 5 and even explored some cool alternative methods. But you might be wondering, *