3 digits

The three-digit, positive integer answer to this question is such that the sum of the factorials of its distinct digits is also the answer to this question.

What is the answer to this question?

The answer is 145.

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2 solutions

Geoff Pilling
Dec 27, 2017

Since 7! > 1000 and the answer only has three digits, you know that the only possible digits are 0-6. Also, since 6! is 720, you can't have a six because this would also imply one of the digits being greater than 6. Which leaves only 0-5. Also, since it is a 3 digit number it must be greater than 99, meaning one of the digits must be a 5, since 2! + 3! +4! < 99. This means the number is 120 + a! + b! where a and b are between 0 and 4 inclusive.

From this, one can quickly deduce that 1 and 4 are the other two digits, making the solution 145 \boxed{145} .

Good solution. I narrowed it down to 1-5, but forgot about the "different digits" part so just guessed it from there

Stephen Mellor - 3 years, 5 months ago

No digit can be 7! or 6!.

So you have 1! to 5! to play with. So quickly do the maths, keeping 1! fixed and get the answer as 145.

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