Here are some nice number facts and tricks you can try out on your friends. They will work without understanding how, but with a little investigation you should be able to figure out how each one works.

#### 1. All four-digit palindromic numbers are divisible by 11.

This is quite easy and nice to prove. Start by writing the palindrome as a sum expressed in terms of the two different digits.

#### 2. Repeat a three-digit number twice, to form a six-digit number. The result will be exactly divisibly by 7, 11 and 13, and dividing by all three will give your original three-digit number.

I found this in a Martin Gardner book. Can you see why it works?

#### 3. Choose a digit from 1-9, and repeat it three times to give a three-digit number. If you divide this number by the sum of its digits, the answer will always be 37.

This is also fun to figure out. It’s an especially nice trick to pull out at a 37th birthday party.

#### 4. If you ask someone to choose ten random digits (not including 0) and multiply them all together, then to tell you all but one of the digits of the answer, you can predict the remaining digit.

Can you guess why this is possible? It relies on the fact that people asked to choose random digits will normally hit enough of the right digits to make this work. It doesn’t work every time, but if you increase the number of digits you ask for, it makes it more likely to work. Hint:^{1}

I’ll post the proofs to all of these next week. Please refrain from posting your own in the comments, so people can try to work them out for themselves.

- The sum of the digits of the number will usually be a multiple of 9. [↩]

111/3 = 37 and 222/6 = 37 etc. Yup! You might want to check your math on that 3rd one. (Posted 8:30 AM UK time on 17 May, 2013)

Hah, well spotted. The proof-reader responsible (me) has been fired out of a cannon (wheee!).

I think in item 3 the target number is 37 not 39.

Yes, spotted and fixed while you were writing that, thanks.

Why did I think it was sufficient just to check the first two? Laziness, that’s why.