**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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