Number Theory
# Factorials

This series of questions helps you understand how to solve the following Level 4 problem about a Diophantine equation:

How many ordered quadruples of positive integers \( (w, x, y, z)\) are there such that \[w! = x! + y! + z! \quad? \]

**Step 1: Understanding the problem - Define the terms**

Which of the following is an ordered quadruple of positive integers that satisfies the condition

\[w! = x! + y! + z! \quad ? \]

**Step 2: Gathering information - Try small cases**

A good way to start is to test small cases to see if we can deduce more information, or even find a solution!

Are there any positive integer solutions to the equation

\[ w! = x! + y! + z! \]

in the case where \( w = 1? \)

**Step 3: Gathering information - Try small cases**

In the previous step, we checked the case where \( w = 1. \) What happens when we work with variables on the right-hand-side? Since the variables are symmetric, it doesn't matter which one we test.

Are there any positive integer solutions when \( z = 2 \) to the equation

\[ w! = x! + y! + z! \quad ? \]

**Step 4: Review - Explain an observation**

Let's review and vocalize what we have learned in steps 2 and 3.

Which variable, when fixed in value, allows us to check finitely many cases to determine if there is a solution to

\[ w ! = x! + y ! + z! \quad ? \]

**Step 5: Develop a plan - Explore the idea**

From step 4, we want to consider what happens when we restrict the value of \( w \). We've already considered the case when \( w = 1 \) is small. Let's consider what happens for larger values of \( w \).

By numerically testing the values, how many positive integer solutions are there to \[ w! = x! + y! + z! \] with \( w = 4 \)?

**Step 6: Review - Explain an observation**

Which of the following is the best explanation for why there are no solutions for \( w = 4 \) to the equation

\[ w! = x! + y! + z! \quad ? \]

**Step 7: Put it together - Integrate the information**

Given \( w \geq 4 \), how many positive integer solutions are there to

\[ w! = x! + y! + z! \]

**Step 8: Put it together - Answer the problem**

How many ordered quadruples of positive integers \( (w, x, y, z)\) are there such that \[w! = x! + y! + z!\quad ?\]

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