Number Riddle
Miss Kate wants to test the mathematical skills of Jake and James. She sets up an integer x for them to guess and gives hints separately to both of them. Both of them are told that:
- 20<x<51
- x = a(mod b) and x=c(mod d)
- Both b and d are different primes larger than 20
- a and c are non-negative integers with a < b and c < d
- Only Jake is told the value of a and b, and only James is told the value of c and d.
She then sets up a meeting for them, but they aren't allowed to tell the others the values of a, b, c, or d. This is what they say:
- Jake: I don't know the number yet
- James: I don't know the number either
- Jake: Now I know the number
- James: I still don't know the number
What is the value of c?
The answer is 21.
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1 solution
The reason Jake cannot find the number is because there are two or more integers n within the defined range that fulfills n=a (mod b). That implies that there are two numbers in the defined range that differs by b. As the smallest integer in the range is 21, and the largest 50, we can tell that b < 50-21+1 = 30. Likewise, d<30. So 20 < b,d <30. That means b and d are among 23 and 29.
If b=29, the only reason Jake doesn't know the number is because a=21, because there are two numbers in the range fulfilling n=21 (mod 29), and there are no other solution for a that causes more than 1 solution. However, if this is the case, Jake knowing that James doesn't know either, implying d=23, won't help at all. After knowing d=23, he can only tell that c ranges from 21-27 to cause James to not know the number because of multiple solutions. Because Jake ends up knowing the number, this isn't the case.
Thus, b= 23. That implies d=29. And because of the reasoning above, the only value of c that can cause James to not know the number is 21.