SOLUTION: I need help solving this problem...looking to find a five digit number: The 3rd and the 5th digits are equal 1st - 3rd = 2nd - 4th 3rd * 4th = 1st ^ 2 - 1 2nd - 1st = sqrt (5

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Question 554273: I need help solving this problem...looking to find a five digit number:
The 3rd and the 5th digits are equal
1st - 3rd = 2nd - 4th
3rd * 4th = 1st ^ 2 - 1
2nd - 1st = sqrt (5th)
Thank you in advance for your help!
Found 2 solutions by Edwin McCravy, KMST:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
There are two such 5-digit numbers, 11000 and 57464

Edwin

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
When faced with a similarly impossible problem that is not answered here, you could give the artofproblemsolving website community a try. The frightfully brainy members of those forums would find an answer if there is one. They used to have a lot of this kind of problem.
MY ANSWER
The two solutions are 11000 and 57464.
THE LONG CASE WORK (Sorry, it's the best I could do).
Let A be the first digit, B the second digit, C the third and fifth digits, and D the fourth digit.
The number would be ABCDC.
We know that A is not zero (or else it would not be a 5 digit number), and that:
A-C=B-D
CD=A%5E2-1
B-A=sqrt%28C%29
The last condition means that C can only be 0, 1, 4, or 9, or else
B-A=sqrt%28C%29 would not be an integer.
It also tells you that B%3E=A to make B-A=sqrt%28C%29%3E=0, and since
A-C=B-D --> D-C=B-A, it also means that D-C%3E=0 and D%3E=C
WITH C=0,
CD=0=A%5E2-1 so A=1
Also B-A=sqrt%28C%29=0, so A=B=1
Then A-C=B-D turns into 1-0=1-D, so D=0.
That would give you the number 11000.
There could be other answers with C=1, C=4, or C=9
CD=A%5E2-1 means that the product CD can only be 0, 3, 8, 15, 24, 35, 48, 63, or 80, for A=1, 2, 3, ...9, respectively. However those values have C (1, 4, or 9) as a factor.
WITH C=1
as a factor, CD=D, and we are limited to 3 of the 9 possibilities above:
CD=D=0 (for A=1), CD=D=3 (for A=2), and CD=D=8 (for A=3).
D=0 does not meet the requirement that D%3E=C
For the other cases, we can calculate B from B-A=sqrt%28C%29 --> B=A%2Bsqrt%28C%29
and see if A-C=B-D is true
For A=2 and D=3, B=A%2Bsqrt%28C%29=2%2Bsqrt%281%29=3
A-C=2-1=1 and B-D=3-3=2 It does not work.
For A=3 and D=8, B=A%2Bsqrt%28C%29=3%2Bsqrt%281%29=4
A-C=3-1=2 and B-D=4-8=-4 It does not work.
WITH C=4 or C=9
The choices for value of the product CD listed above narrow to
CD=8=4%2A2 (for C=4, A=3) with D=2,
CD=24=4%2A6 (for C=4, A=5) with D=6,
CD=63=9%2A7 (for C=9, A=8) with D=7
The first choice (C=4, A=3, D=2) and the last one (C=9, A=8, D=7) are eliminated because they do not comply with D%3E=C.
For the remaining possibility, we calculate B from B-A=sqrt%28C%29 --> B=A%2Bsqrt%28C%29
and see if A-C=B-D is true.
For C=4, A=5, D=6 --> B=A%2Bsqrt%28C%29=5%2Bsqrt%284%29=5%2B2=7
Then A-C=5-4=1 and B-D=7-6=1 We have another solution (57464).