Number Theory

Factorials

Number Theory Step Up to Level 4 - Set 1

               

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) (w, x, y, z) are there such that w!=x!+y!+z!?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!?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! w! = x! + y! + z!

in the case where w=1? w = 1?

Step 3: Gathering information - Try small cases

In the previous step, we checked the case where w=1. 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 z = 2 to the equation

w!=x!+y!+z!? 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!? 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 w . We've already considered the case when w=1 w = 1 is small. Let's consider what happens for larger values of w w .

By numerically testing the values, how many positive integer solutions are there to w!=x!+y!+z! w! = x! + y! + z! with w=4 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 w = 4 to the equation

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

Step 7: Put it together - Integrate the information

Given w4 w \geq 4 , how many positive integer solutions are there to

w!=x!+y!+z! w! = x! + y! + z!

Step 8: Put it together - Answer the problem

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

×

Problem Loading...

Note Loading...

Set Loading...