Simplify 84! / (82! + 83!)

Challenge

\frac{84 !}{82 ! + 83 !}

Press play — the board writes itself.

0:00 / 2:06

Transcript

84! divided by 82! + 83!. The top number alone has 127 digits — your calculator just gives up. We're going to collapse the whole thing in three moves. Watch.

$\frac{84 !}{82 ! + 83 !}$

Everybody knows one factorial trick: 84! over 83! is just 84. So it's tempting to cancel something here and call it done. Careful — the bottom isn't one factorial. It's a SUM. And you never, ever cancel across a sum.

$\frac{84 !}{83 !} = 84$

Here's the key instead. 83! is just 83 times 82! — every factorial carries the smaller ones inside it. Which means 82! is hiding in BOTH terms downstairs.

$83! = 83 \cdot 82!$

So pull it out — factor it, exactly like pulling an x out of a polynomial. What's left inside is 1 + 83… which is 84. Look at that: the plus sign is GONE. The bottom is a clean product now.

$82! + 83! = 82! (1 + 83) = 82! \cdot 84$

Open the top just enough to match: 84! is 84, times 83, times 82!. Now everything monstrous cancels — the 84s go, the 82!s go… and all that survives is 83.

$\frac{84 \cdot 83 \cdot 82 !}{84 \cdot 82 !} = 83$

A 127-digit number, divided by another one nearly as big — and it comes out exactly 83. Not roughly. Exactly.

The move to keep: when terms are ADDED, factor out what they share before you compute anything. It's the humble ab-plus-ac trick from algebra — and it just took down two factorials.

The monster that collapses · 2:06

The factorials look terrifying — $84!$ alone runs to 127 digits — so the whole game is to make sure you never actually compute one. The key is that $82!$ is hiding inside every term.

Factor it out of the denominator. Since $83! = 83 \times 82!$ ,

$82! + 83! = 82!\,(1 + 83) = 82! \times 84$

Now open the numerator just enough to expose the same piece: $84! = 84 \times 83 \times 82!$ . Everything monstrous cancels:

$\frac{84!}{82! + 83!} = \frac{84 \times 83 \times 82!}{84 \times 82!} = 83$

A 127-digit number divided by one nearly as vast, and the answer is a tidy $83$ .

What this hides

Big numbers feel like they demand big computation, so the instinct is either to reach for a calculator — which overflows — or to glance at $\tfrac{84!}{83!} = 84$ and call it done, forgetting the extra $82!$ waiting downstairs. But the denominator is a sum, and a sum has to be factored before a single thing cancels.

The transferable idea is the humble one behind $ab + ac = a(b + c)$ : when terms are added, pull out what they share before you evaluate. Here $82!$ simply plays the role of $a$ . That one move turns an impossible arithmetic problem into a one-line cancellation — the same reason factoring earns its keep in algebra.