Question

N! + (n + 1)! = 144 ​

Answer 1

We can factor out (n!) from the left side of the equation:

n! + (n + 1)! = n!(1 + n + 1) = n!(n + 2)

So we have:

n!(n + 2) = 144

We can find the prime factorization of 144:

144 = 2^4 * 3^2

So we need to find two factors of 144 that differ by 2. We can see that 12 and 14 work:

12! = 479001600

13! = 6227020800

14! = 87178291200

So n = 12, and we can check that:

12! + 13! = 479001600 + 6227020800 = 6706022400

13! + 14! = 6227020800 + 87178291200 = 93405312000

So 12! + 13! is the only solution to the equation.