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Gift · May 8, 2007, 01:59 PM

I really have no idea what to do with this question. So I hope you could help me.

"the product of five prime positive integers is a six-digit number, where all of the digits are the same. Find the six-digit number described above. "

Capuchin · May 8, 2007, 02:13 PM

That's actually rather hard, however it's fairly easy to guess at the answer with weak logic if you don't know the answer.

You have to choose between 111111, 222222... 999999, so a choice of 9 numbers.

Most of these (222222, 333333, 555555, 777777, 999999) are just 111111 with an extra prime factor. The others (888888, 444444, 666666) are just 222222 with extra prime factors. So you can argue that 111111 definitely has the least number of prime factors. But does it have 5? Or less? So, that's the one to try first. Do you know how to find prime factors? Trial and error with a little bit of logic shouldn't be too hard.

Gift · May 8, 2007, 02:20 PM

Thank you so much!! I'm not really sure how to find which one's a prime number. But I try to divide the number ,if there are only itself and 1 that are the factors. I just say that it's a prime number :]

Capuchin · May 8, 2007, 02:25 PM

Well, Although I know what the answer is, you need to convince yourself if 111111 has 5 factors. It might only have 4 factors, then one of the other ones will have 5 factors.

Can you work out what the prime factors of 111111 are?

Gift · May 8, 2007, 02:26 PM

Okay... thank you so much!

Nosnosna · May 8, 2007, 02:40 PM

I'm trying to figure out how to solve this without actually factoring. I've got it narrowed down to two possibilities very quickly, but I can't eliminate the last one quite yet.

All in all, a nice bit of mental floss.

Edit: Got a pure logic way, but it's somewhat inelegant at the end... breaking out the old factoring tricks. It also relies on the assumption that there's one and only one correct answer.