All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
A special short-list (Posted on 2015-12-08) Difficulty: 3 of 5
Some integers are one more than the sum of the squares of their digits in base 10.

Prove that the set of such integers is finite and list all of them.

See The Solution Submitted by Ady TZIDON    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Analytical solution Comment 3 of 3 |
Even though brute force is pretty quick I thought this was interesting because it shows the connection between the two solutions.

Any solution is clearly positive.
No solution can have 1 digit since no real is one less than its square.

As noted if we try 3 digits 9^2+9^2+9^2+1=244 so the first digit is 2 or less.  
But then 2^2+9^2+9^2+1=167 so the first digit is 1 and the second is 6 or less.
But then 1^2+6^2+9^2+1=119 so the second digit is 1 or less.
But then 1^1+1^2+9^2=83 so there can be no 3 digit answer.

Any solution must have 2 digits.
Let the digits be A and B so we have
10A + B = A^2 + B^2 + 1
Solving for A by quadratic formula simplifies to:
The only digit for B that makes the sqrt rational (a bit of brute force here) is B=5 which gives 2.
A=5+/-2 = 7 or 3
So the numbers are 75 and 35.

  Posted by Jer on 2015-12-08 11:25:09
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (2)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (1)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2020 by Animus Pactum Consulting. All rights reserved. Privacy Information