Nothing to do with binary, folks!

[Paul Howe]

Here's a puzzle anyone who's done GCSE maths is equipped to do.

What is the smallest number (apart from 0), consisting entirely of 0's and 1's, that is divisible by 225?

No points will be given for just using a computer to find the answer!

[Ben Pugh]

Is it 11111111100?

[Paul Howe]

I'm neither going to confirm or deny answers without at least some explanation as to how you came up with it!

[Ben Wilson]

Piece of piss to work out, surely- 225= 15^2= 3^2 * 5^2, therefore the number has to be divisible by 9 and by 25. The first criterion means its digits must sum to a multiple of 9, and as we're using only 1s and 0s, it must therefore contain 9 1s. It must also be divisble by 25, meaning it must be a multiple of 100- as it can't end -25, -50 or -75. Hence, we get 11 111 111 100.

[Ben Pugh]

225 - 2 + 2 + 5 = 9.

225 and the answer to your problem are both divisible by 9. So if the number is just made up of 1's and 0's, there have to be at least 9 1s. I just stuck 9 1s in a row and put a 0 after them, 1111111110 is not divisible by 225, 11111111100 is so I guessed that that was the answer.

[Paul Howe]

Nice, very good guys. I've clearly gone from too hard to too easy, next time I shall try and find a happy medium.

Ben, I'd say it was trivial if you've seen this kind of problem before, but might cause a few headaches for those who haven't. I just thought it'd be a nice puzzle for Countdowners because it uses Carol's nines trick!

[Joseph Bolas]

Nice, very good guys. I've clearly gone from too hard to too easy, next time I shall try and find a happy medium.

Ben, I'd say it was trivial if you've seen this kind of problem before, but might cause a few headaches for those who haven't. I just thought it'd be a nice puzzle for Countdowners because it uses Carol's nines trick!

I'm glad someone knew this answer, I don't think I would've worked out the logic that the number had to be divisible by both 225 and 9 (its so obvious I would've missed it).

[Kirk Bevins]

I really like this problem, Paul. Makes a change from some of the other ones you've posted with ridiculous trick answers! This really gets people thinking. I might use it in class one day at school.

[Charlie Reams]

I might use it in class one day at school.

Did you ever do this?

[Kirk Bevins]

I might use it in class one day at school.

Did you ever do this?

No! I'd forgotten all about it. I'll have to try and remember it.

[Martin Gardner]

Interesting that I got my dad the 'Who Wants To Be A Millionaire' for Xmas (I never know what to get him) and I read all the million pound questions, and I could get two right for certain and only guess the rest (or take the money, in the figurative sense). The easy one was 'What number does 10 represent in binary?' which if you know the answer is really easy.

Also I was thinking that the multiple of nine rule works for any numbers system, does it? I.e. for base X it works for multiples of X-1. Amusing this always works for binary, as every number is divisible by 1!

[Paul Howe]

Interesting that I got my dad the 'Who Wants To Be A Millionaire' for Xmas (I never know what to get him) and I read all the million pound questions, and I could get two right for certain and only guess the rest (or take the money, in the figurative sense). The easy one was 'What number does 10 represent in binary?' which if you know the answer is really easy.

Indeed. Are you sure it wasn't the 64 pound question?

Also I was thinking that the multiple of nine rule works for any numbers system, does it? I.e. for base X it works for multiples of X-1. Amusing this always works for binary, as every number is divisible by 1!

Yeah, that's right.

[Jon Corby]

Indeed. Are you sure it wasn't the 64 pound question?

Lolz!

[Charlie Reams]

The easy one was 'What number does 10 represent in binary?' which if you know the answer is really easy.

Because, like, most questions are really hard if you know the answer.

[Adam Dexter]

Am I being dense, or is it 1050? 1050 / 4 = 225 does it not? :S

[Junaid Mubeen]

Am I being dense, or is it 1050? 1050 / 4 = 225 does it not? :S

[Charlie Reams]

Am I being dense, or is it 1050? 1050 / 4 = 225 does it not? :S

Spot which of the following numbers is not a 0 or a 1:
1, 0, 5, 0.

[Gavin Chipper]

Am I being dense, or is it 1050? 1050 / 4 = 225 does it not? :S

Spot which of the following numbers is not a 0 or a 1:
1, 0, 5, 0.

The second 0.

[Kai Laddiman]

Am I being dense, or is it 1050? 1050 / 4 = 225 does it not? :S

Spot which of the following numbers is not a 0 or a 1:
1, 0, 5, 0.

Ummm...errr...any clues?

[Adam Dexter]

Oh dear.... :$

More to the point... 225 * 4 = 900... :$

[Martin Gardner]

I've always wanted to proove the multiple-of-nine rule, but I've never been able to do it! Anyone got one? I suppose Google could probably find it pretty quickly...

[Peter Mabey]

As dividing 10 by 9 leaves remainder 1, any digit followed by any number of 0s leaves that digit as remainder on dividing by 9. Write your number as a sum like that, and you're done. (e.g. 486 -> 400+80+6, so remainder is same as that of 4+8+6=18 - obviously 0)
As the sum of digits of a multi-digit number is always less than that number, you can always repeat the process till you end with a single digit which will be 9 only if you had a multiple of 9 to start with.