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My numbers (Posted on 2014-11-14) Difficulty: 2 of 5
Find my two numbers:

1. Both are positive integers..
2. Their difference is a prime.
3. Their product is a perfect square.
4. Their sum's last digit is 3.

Rem: It was solved by me immediately upon presentation by a friend.

I leave it to you to check whether there is more than one solution.


See The Solution Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: All of the solutions | Comment 2 of 3 |
(In reply to All of the solutions by Jer)


Nicely done, but your proof can be simplified further, as follows:

From 4.  The numbers must be an even and an odd.
So to fit 3.  The numbers themselves must be perfect squares.
All even perfect squares end in 0, 4, or 6.
All odd perfect squares end in 1, 5, or 9.
The only combination whose sum ends in 3 is 4+9.

Since the squares end in 4 and 9, their difference must end in 5.
The only prime that ends in 5 is 5 itself.
The only squares that are 5 apart are 4 and 9.
Hence, this is the only solution.

  Posted by Steve Herman on 2014-11-14 13:59:10
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