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1-2-3 Remainder Resolution (Posted on 2016-02-16) Difficulty: 3 of 5
Abe challenges Bee to determine a 3-digit positive integer N.
It is known that the number formed by the last two digits of N when divided by 9, yields a remainder of 3.

Abe makes the following statements, precisely one of which is false:
  1. N divided separately by each of 2, 4, 6, and 8 yields a remainder of 1.
  2. N divided separately by each of 5 and 7 yields a remainder of 2.
  3. N divided separately by each of 5 and 11 yields a remainder of 3.
Determine the value of N from the above statements and given clues.

No Solution Yet Submitted by K Sengupta    
Rating: 4.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution d3 answer | Comment 2 of 12 |
1) Statements 2 and 3 are contradictory, so 1 is true.  Therefore, N = 24x + 1

2) From statement 0, N = 100y + 9z + 3.  

3) If statement 3 is true, then (100y + 9z +3) mod 11 = 3 = (100y + 9z + 3) mod 5
   Then 9z = 0 mod 5 and y = 2z mod 11.
   But 9z = 0 mod 5 implies z = 0 or 5, and this fails if y = 2z mod 11.
   So statement 3 is false.

4) So statement 2 is true.  Then (100y + 9z +3) mod 5 = 2
   So z = 1 mod 5.  z = 1 or 6. 
     If z = 1, then N is even, and cannot be 24x + 1
     So z must be 6.  
       Then 100y + 57 = 1 mod 24.  
       4y = 16 mod 24
       y = 4
       
So the final answer is 457.

Checking, 457 mod 7 = 2

Edited on February 16, 2016, 11:56 am
  Posted by Steve Herman on 2016-02-16 11:55:49

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