Find Natural Numbers X, Y, Z: Xyz / Yz = 51 (Remainder 0)

by SLV Team 58 views

Hey guys! Today, we're diving into a fun math problem where we need to figure out some natural numbers. Specifically, we're looking for natural numbers x, y, and z, with a couple of rules: x and y can't be zero, and when you divide the number 'xyz' by 'yz', you get 51 with no remainder. Sounds intriguing, right? Let's break it down and see how we can crack this.

Understanding the Problem

So, what exactly are we trying to do? We need to find numbers x, y, and z that fit a particular condition. Remember, natural numbers are the positive whole numbers (1, 2, 3, and so on). The number 'xyz' isn't just x times y times z; it represents a three-digit number where x is the hundreds digit, y is the tens digit, and z is the units digit. Similarly, 'yz' represents a two-digit number. The problem states that when we divide the three-digit number 'xyz' by the two-digit number 'yz,' we get exactly 51. No remainders allowed!

This is super important because it means that 'xyz' is perfectly divisible by 'yz'. It's like saying 102 divided by 2 is 51, no leftover bits. To make this clearer, we can express the three-digit number 'xyz' and the two-digit number 'yz' in their expanded forms. This will help us understand the relationship between the digits and the numbers they represent. Let's dive into that next!

Breaking Down the Numbers

To really understand what's going on, let's express our numbers in their expanded forms. This will make the math a lot clearer. The three-digit number 'xyz' can be written as 100 * x + 10 * y + z. Think about it: the digit x is in the hundreds place, so it's worth 100 times its value; y is in the tens place, so it's worth 10 times its value; and z is in the units place, so it's just worth its face value. Similarly, the two-digit number 'yz' can be written as 10 * y + z. Now, we know that xyz / yz = 51, which we can rewrite as:

(100 * x + 10 * y + z) / (10 * y + z) = 51

This equation is the key to solving our problem. It shows us the exact relationship we need to work with. To solve for x, y, and z, we need to manipulate this equation. The next step is to get rid of the fraction, which will make things much easier to handle.

Eliminating the Fraction

To get rid of the fraction in our equation, we simply multiply both sides by the denominator, which is (10 * y + z). This gives us:

100 * x + 10 * y + z = 51 * (10 * y + z)

Now, we need to distribute the 51 on the right side of the equation. This means multiplying 51 by both 10 * y and z:

100 * x + 10 * y + z = 510 * y + 51 * z

This equation looks a bit more complex, but it's actually a big step forward. We've eliminated the fraction, and now we have a linear equation with three variables (x, y, and z). The next step is to simplify this equation by grouping similar terms together. This will help us isolate the variables and make it easier to solve for them.

Simplifying the Equation

Okay, let's simplify the equation we got in the last step. We have:

100 * x + 10 * y + z = 510 * y + 51 * z

Our goal is to isolate the variables, so let's move the terms involving y and z from the left side to the right side of the equation. We do this by subtracting 10 * y and z from both sides:

100 * x = 510 * y + 51 * z - 10 * y - z

Now, we can combine the like terms on the right side:

100 * x = 500 * y + 50 * z

This looks much better! Notice that all the terms are multiples of 50. This suggests we can simplify the equation further by dividing both sides by 50:

2 * x = 10 * y + z

This is a significant simplification. We've reduced the equation to a much more manageable form. This simplified equation gives us a direct relationship between x, y, and z. Now, we can start thinking about the possible values for these variables, keeping in mind that they are natural numbers and x and y cannot be zero. Let's explore the possible values in the next section.

Finding Possible Values

Now that we have the simplified equation 2 * x = 10 * y + z, we can start figuring out possible values for x, y, and z. Remember, x, y, and z are natural numbers (1, 2, 3, ...) and x and y can't be zero. This is crucial because it limits the range of possible values. Let's analyze the equation:

  • The right side of the equation, 10 * y + z, represents a two-digit number (since y and z are digits). This is important because it tells us the range of possible values for 2 * x.
  • Since y and z are digits, the smallest possible value for 10 * y + z is 10 (when y = 1 and z = 0), and the largest possible value is 99 (when y = 9 and z = 9).
  • Therefore, 2 * x must be between 10 and 99. This means that x must be between 5 and 49.5. However, since x is a natural number, it must be between 5 and 49.

Now, let's consider the possible values for y. Since 10 * y is a significant part of the right side of the equation, it makes sense to start there. Let's go through the possibilities:

Analyzing Possible Values for y

We know that 2 * x = 10 * y + z, and x, y, and z are natural numbers with x and y not equal to zero. Let's think about the possible values for y. Since y is in the tens place of a two-digit number, it can be any digit from 1 to 9. Let's analyze how different values of y affect the equation:

  • If y = 1, the equation becomes 2 * x = 10 + z. This means that 2 * x must be a number between 10 and 19 (since z can be any digit from 0 to 9). Therefore, x can be any number between 5 and 9.
  • If y = 2, the equation becomes 2 * x = 20 + z. Now, 2 * x must be between 20 and 29, so x can be any number between 10 and 14.
  • We can continue this pattern for all values of y from 1 to 9. For each value of y, we get a range of possible values for x. Once we have a value for x and y, we can calculate z using the equation z = 2 * x - 10 * y.

This method allows us to systematically find all possible solutions. However, we need to remember that z must also be a natural number between 0 and 9. This constraint will help us narrow down the solutions. Let's look at a few examples to see how this works in practice.

Examples of Finding Solutions

Let's take a couple of examples to illustrate how we can find solutions. Remember our equation: 2 * x = 10 * y + z.

  • Example 1: Let's say y = 5.
    • The equation becomes 2 * x = 50 + z.
    • Since z can be a digit between 0 and 9, 2 * x must be between 50 and 59.
    • This means x can be any number between 25 and 29.5. Since x must be a natural number, the possible values for x are 25, 26, 27, 28, and 29.
    • Now, let's check each value of x:
      • If x = 25, then z = 2 * 25 - 50 = 0. So, we have a solution: x = 25, y = 5, z = 0.
      • If x = 26, then z = 2 * 26 - 50 = 2. So, we have another solution: x = 26, y = 5, z = 2.
      • We can continue this process for x = 27, 28, and 29.
  • Example 2: Let's say y = 9.
    • The equation becomes 2 * x = 90 + z.
    • Since z can be between 0 and 9, 2 * x must be between 90 and 99.
    • This means x can be any number between 45 and 49.5. The possible values for x are 45, 46, 47, 48, and 49.
    • Again, we check each value of x:
      • If x = 45, then z = 2 * 45 - 90 = 0. So, we have a solution: x = 45, y = 9, z = 0.
      • If x = 46, then z = 2 * 46 - 90 = 2. So, we have another solution: x = 46, y = 9, z = 2.
      • We continue this for the remaining values of x.

By going through each possible value of y and then checking the corresponding values of x, we can systematically find all the solutions to our problem. This might seem a bit tedious, but it's a reliable way to ensure we don't miss any solutions. Now, let's try to summarize our findings and see what the general solution looks like.

Summarizing the Solutions

After going through the process of analyzing possible values for y and calculating corresponding values for x and z, we can summarize our solutions. Remember, we're looking for natural numbers x, y, and z (with x and y not being zero) that satisfy the equation 2 * x = 10 * y + z.

By trying different values for y (from 1 to 9) and then finding the corresponding values for x and z, we can generate a set of solutions. It's important to remember that z must be a digit between 0 and 9, which limits the possible solutions. Here are a few solutions we found earlier:

  • x = 25, y = 5, z = 0
  • x = 26, y = 5, z = 2
  • x = 45, y = 9, z = 0
  • x = 46, y = 9, z = 2

If we continue this process for all values of y, we'll find a complete set of solutions. The solutions will be in the form of (x, y, z) ordered triples. These ordered triples represent the digits of our numbers 'xyz' and 'yz' that satisfy the original condition: xyz / yz = 51 with a remainder of 0.

To be completely sure we've found all the solutions, it's a good idea to organize our work and check each possible combination. We can do this by creating a table or a list where we systematically record the values of x, y, and z that work. This helps us avoid missing any solutions and ensures we have a complete answer to the problem. So, guys, this is how we tackle this kind of number puzzle! Remember, breaking down the problem, simplifying the equations, and systematically checking the possibilities is the key to success. Keep practicing, and you'll become a pro at solving these types of problems!