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Chinatown's problem (Posted on 2015-07-02) Difficulty: 2 of 5
x ≡ 0 mod 2
x ≡ 1 mod 3
x ≡ 2 mod 5
x ≡ 3 mod 7

a. What is the smallest x?
b. What is the largest 4-digit x?

No Solution Yet Submitted by Ady TZIDON    
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Solution solution with some explanation | Comment 2 of 3 |
a.

x is even, 2 + 2k
x is 1 mod 3, so 4 + 6k
x is 2 mod 5, so 22 + 30k
x is 3 mod 7:
   22 is 1 mod 7
   30 is 2 mod 7
   
   answer is 52
 

                        
b.

10000 is 
   0 mod 2
   1 mod 3
   0 mod 5
   4 mod 7
   
we need to subtract y

y is even, 2 + 2k
y is 0 mod 3, so 6 + 6k
y is 3 mod 5, so 18 + 30k
y is 1 mod 7:
   18 is 4 mod 7
   30 is 2 mod 7 so k is 2
   
   answer is 10000 - (18 + 30*2)
           = 10000 - 78 = 9922
           
Note: in each of these, of course, successive k's are not the same, but merely indicate separation of possible successive values.
           

Edited on July 2, 2015, 9:22 am
  Posted by Charlie on 2015-07-02 09:19:53

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