I am thinking of a 5 digit number that when tripled is a perfect square.

Also, when the 5 digit number is split, the first number is double the second one. What is the five digit number?

(Splitting the 5 digit number into two numbers means 12345 into 1 and 2345 or 123 and 45.)

The number is 13467. 67*2 = 134, and 13467*3 = 40401 = 201*201

This is how I found the number:
The splitting the number in two requirement means the number had to start with a 1. The largest value the second half could have is 99, and 2*99 = 198. So the number has to be between 10050 and 19899. Also, the digit in the hundreds place must be an even number, since it comes from another number being multiplied by 2.

Tripling those numbers gives a range of 30150 to 59697. The square roots of those numbers are 173.64 and 244.33. So that means a number between 174 and 244 that gets squared and divided by three.

Since the number that is squared gets divided by 3, that means the starting number must be a multiple of 3, so it must be one of 174, 177, 180, 183, ..., 237, 240, 243; a list of 24 numbers.

At this point I decided to just do trial-and-error, since 24 is a reasonably small number of iterations, and found the above answer.

Posted by Ender on 2003-08-20 14:49:53