A Nice Factorial Equation

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[wtspman]

You can save a few steps at the start by multiplying the first term on the left hand side by n/n to make the common denominator: (n – 1)! x n = n!. Once you’ve done that the next step takes you directly to (n – 1)/n! = 1/8.

[MushookieMan]

You said n=1 isn't allowed because one of the denominators was (n-1)!, but that's well defined as 1. n=1 isn't a solution by manual checking.

[matthewfeig]

This problem is a subcase of a much more general diophantine equation: 1/a - 1/b = 1/8 with a, b>0.

And this equation only has three solutions in positive integers: (a, b) =(4, 8), (6, 24), (7, 56). (If a>=8, then b is undefined or negative, so just check a=1, 2, 3, 4, 5, 6, 7.)

So this problem doesn't really have much to do with the specific formulas a=(n-1)! and b=n!. They just narrow us down from three possibilities to one.

[GlorifiedTruth]

I love this channel not just for the math but for the heart.

BTW, I thought I kicked his ass on this, coming up with an answer in about 30 seconds. But now I see I left off a factorial sign, which made it WAY easier and yielded a wrong answer to boot. (I was so pleased with myself I didn't even check to see if my answer worked.)

[hemaldoshi]

n(n-2)! = 8
8 = 1×8, 2×4, 4×2, 8×1
Since n doesn't equal 1
And n > n-2
So the possible solution is 4×2
Therefore, n(n-2)! = 4×2
And 2 =2!
So n = 4

[xakdog]

Could be simpler 1/(n - 1)! same as n/n! Hence 8(n-1) = n! then it's super easy to guess - factorial grows faster than the product on the left side

[mohitg]

At 3:40, you could have tried in way like at the RHS=8 (n-1)! as 4*3*2 and compare with the LHS and you easily get that n=4 is a Solution.

[weylguy]

I used the fact that n!/(n-1)! = n, but still had to guess that n = 4.

[piotrskalski]

A quick manipulation gave n!=8(n-1), from where I was able to guess that n=4 is a solution.

[xchomphk.]

1/(n!/n) - 1/n! = 1/8
n/n!-1/n! =1/8
8n-8 = n!. just trial and error until you find a value for n, n=4

[carlyet]

n == 4 just use algebra (n-1)! is a factor of n! {n.(n-1)!} [n-1]/n! === 1/n.(n-2)!
giving 8 = n.(n-2)!

[omar.ma]

Did it simplier
The common denominator is n!

=(n-1)/1...(n-2)(n-1)n=1/8
as 1 is not a solution by checking it
Simplify by n-1
We get 1.2...(n-2).n=8
n<=4 because 5!.3!.2!.1!=30
And checking for n=2, 3, 4

[alextang]

It is not difficult as the number is not big. 😋😋😋😋😋😋