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# TI-84+ and Casio Micropython (fx-CG50): Solving Equations Involving Factorials (Inverse Factorial)

TI-84+ and Casio Micropython (fx-CG50):  Inverse Factorial of Integer

Introduction

The goal is to solve the equation for n:

x = n!

One way to do this is to use a variation of the gamma function which invovles an improper integral:

x = ʃ (t^ n * e^(-n) dt, 0, ∞)

(note that n! = gamma(n+1) )

Or use an approximation formula.

Another approach is to use an iterative method, which does not use calculus.  Hence, if x and n are integers then x can be written as:

x = n! + r

Where r is a remainder.  This method involves successive division.

Loop:

1.  Set n = 2  (since 1! = 1) and set t = x.  The  variable t will be working copy of x.
2.  Divide t by n and store in t, t = t / n
3.  Increase n by 1, n = n + 1
4.  If t is less than or greater than n, repeat steps 2 and 3.
5.  If t = n, then x = n!.  Done.
6.  If t ≠ 1, then do the following.  Set n = n – 1 and r = x – n!.  The answer is x = n! + r.

Example 1:    120 = n!

n = 2, t = 120/2 = 60,  60 > 3
n = 3,  t = 60/3 = 20,  20 > 4
n = 4,  t = 20/4 = 5, 5 = 5

Since 5 = 5, 120 = 5!

Example 2:   177 = n!

n = 2, t = 177/2,  88.5 > 3
n = 3, t = 88.5/3, 29.5 > 4
n = 4,  t = 29.5./4, 7.375 > 5
n = 5, t = 7.375/5, 1.475 < 6.  Stop

r = 177 – 5!  = 57

Hence:  177 = 5! + 57

TI-84 Plus Program INVFACT

“EWS 2018-11-11”
Disp “X = N! + R”
Input “X: “, X
X→T
2→N
Repeat T≤N
T/N→T
N+1→N
End
If X-N!=0
Then
Disp N, “!”
Else
N-1→N
X-N!→R
Disp N, “! +”, R
End

Casio Micropython (fx-CG 50) Script invfact.py

import math
print(“x = n! + r”)
x=float(input(“x: “))
t=x
n=2
while t>n:
t=t/n
n=n+1
f=1
for i in range (1, n+1):
f=f*i
if x-f==0:
print(n,”!”)
else:
n=n-1
f=f/i
r=int(x-f)
print(n, “! +”,r)

Examples:

24 = 4!

26 = 4! + 2

53 = 4! + 29

Eddie

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