39x + 41y + 42z = 456. x, y, z are natural  numbers. Find x + y + z. Don't guess numbers

To solve the equation \(39x + 41y + 42z = 456\) where \(x\), \(y\), and \(z\) are natural numbers, we will approach it systematically without guessing.

Given:
\[
39x + 41y + 42z = 456
\]

### Step 1: Express \(z\) in terms of \(x\) and \(y\)
We can rearrange the equation to solve for \(z\):
\[
42z = 456 - 39x - 41y
\]
\[
z = \frac{456 - 39x - 41y}{42}
\]
Since \(z\) is a natural number, the expression \(\frac{456 - 39x - 41y}{42}\) must be a positive integer.

### Step 2: Ensure \(456 - 39x - 41y\) is divisible by 42
For \(z\) to be a natural number, \(456 - 39x - 41y\) must be divisible by 42. We need to find values of \(x\) and \(y\) such that this divisibility condition holds.

We will evaluate the expression modulo 42 to identify constraints on \(x\) and \(y\):

\[
456 \equiv 36 \ (\text{mod}\ 42)
\]
\[
39x \equiv 39x \ (\text{mod}\ 42)
\]
\[
41y \equiv -y \ (\text{mod}\ 42)
\]
So,
\[
36 - 39x - y \equiv 0 \ (\text{mod}\ 42)
\]
\[
39x + y \equiv 36 \ (\text{mod}\ 42)
\]

### Step 3: Simplify \(39x + y \equiv 36 \ (\text{mod}\ 42)\)
We need to solve the congruence equation \(39x + y \equiv 36 \ (\text{mod}\ 42)\).

Since \(39x \equiv -3x \ (\text{mod}\ 42)\), we have:
\[
-3x + y \equiv 36 \ (\text{mod}\ 42)
\]
\[
y \equiv 3x + 36 \ (\text{mod}\ 42)
\]

### Step 4: Plug in natural numbers for \(x\) and solve for \(y\)
Since \(x\), \(y\), and \(z\) must be natural numbers, let's find values of \(x\) such that \(y = 3x + 36\) is also a natural number and solves the original equation.

For \(x = 2\):
\[
y = 3(2) + 36 = 42
\]

Substitute \(x = 2\) and \(y = 42\) into the original equation:
\[
39(2) + 41(42) + 42z = 456
\]
\[
78 + 1722 + 42z = 456
\]
This gives:
\[
42z = 456 - 1800 = -1344
\]
Since \(z\) cannot be negative, we continue to test other values of \(x\).

For \(x = 4\):
\[
y = 3(4) + 36 = 48
\]
Substitute \(x = 4\) and \(y = 48\) into the original equation:
\[
39(4) + 41(48) + 42z = 456
\]
\[
156 + 1968 + 42z = 456
\]
This gives:
\[
42z = 456 - 2124 = -1668
\]
Again, \(z\) cannot be negative. Let's proceed until we find a natural \(z\).

Continue this process until a valid combination is found, verifying each step. Finally, check the consistency with the given conditions. Due to the complexity, a more advanced method like the extended Euclidean algorithm might be used for efficient solution-finding, or computational tools might assist in solving. Would you like to proceed with that?

### Relative Questions:
1. Can we simplify the modulus calculation further?
2. What is the significance of ensuring divisibility by 42?
3. How do extended algorithms simplify solving linear Diophantine equations?
4. Why is it necessary to test different values of \(x\) and \(y\) systematically?
5. What computational tools can assist in solving such equations more efficiently?

### Tip:
Always double-check the steps in modular arithmetic to ensure correct simplification when dealing with equations involving large numbers and constraints.

Explore the detailed solution to the linear Diophantine equation 39x + 41y + 42z = 456, focusing on natural number solutions for x, y, and z. Learn about modular arithmetic and the extended Euclidean algorithm. Suitable for advanced mathematics enthusiasts.

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