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.)

13467

For a 5-digit number to be 'split' into two parts, one of which is double the other, the parts must of course have three and two digits, respectively.

Let x represent the two-digit number (which must be between 50 and 99). 2x is then the three-digit number. To make the five-digit number, these three digits must be moved two places to the left, so we multiply by 100.

Therefore, the five-digit number is represented by 201x, where x is the two-digit number in question. If we triple this number, we have a perfect square, equal to 603x.

The prime factorization of 201 is just 3×67, so we have that 3×3×67×x is a perfect square. There are two ways for this to happen; x must be equal to 3×3×67, so that the resulting number is 603², or x must equal 67=603, so that the resulting number is (3×67)(3×67)=201². We already know that x has only two digits, so the latter is the case.

Therefore, the five digit number is 13467, which is equal to 201².

Posted by DJ on 2003-08-20 16:31:07