Lou Zerr is trying to pick up a woman at a bar. The woman, Denise, is reluctant to give out her number, but she gives Lou a clue.

"My sister, Elise and I have the same digits in our phone numbers, just in reverse order. Elise's number is also evenly divisible by mine."

What is Denise's phone number?

(Note: this IS NOT a real exchange, as far as I know. Assume the phone number has 10 digits: 7 digit number + 3 digit area code.)

To solve this, I need to figure how many times Denise's number goes into Elise's.  First I assume that Denise's number starts with a one.

1?????????
x        ?
?????????1

The multiplier must be an odd number, because Elise's is an odd number.  It can't be 1, or they would have the same number (the problem says "phone numbers").  It can't be 5 either.  3 is also impossible because it would make Elise's first digit 7.  Therefore (if Denise's first digit is 1) the multiplier is 9.  I think that a multiplier of 9 is unlikely to work.

So now I will assume Denise's first digit to be 2.  I think the lowest possible multiplier is most likely to work.  2 and 3 won't work because the digits on the right and left sides wouldn't come out right.  So I will try 4.

21??????78  There is nothing carried into the last
x        4  column, so D.'s second number is 0, 1, or
87??????12  2. Only 1 works, because E.'s 9th digit must be odd.

For the middle 6 digits:
We know a 3 is carried into the rightmost digit, and a 3 must be carried out of the left-most digit.
A quick check reveals that 9 works.  The same follows for the rest of the digits.

2199999978
x              4
8799999912

A calculator check shows that this is true.  So Denise's number is (219) 999-9978.

However, I haven't proved uniqueness.

Posted by Tristan on 2004-04-18 12:01:19