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Part A:

For the two systems of linear congruences, one system has integer solutions while the other does not. For the system with integer solutions, write down 2 of them whose difference is less than 192. For the other system, explain why no integer solution exists.

    A: n congruent 13 (mod 16)
       n congruent 5 (mod 12)

    B: n congruent 14 (mod 16)
       n congreunt 4 (mod 12)

Part B:

Let a1 and a2 be integers. 
Let m1 and m2 be natural numbers.
Let d = gcd(m1,m2)
Based on your observations from part A, complete the following proposition and prove it.

Proposition1: The system:
n congruent a1 (mod m1) 
n congruent a2 (mod m2)
has an integer solution if and only if ____________
(The blank needs to be filled with a simple condition on a1,a2,d)

Any tips would be great! Thanks in advance!

Benjamin Gruenbaum
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user3422146
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1 Answers1

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16 and 12 are not coprime, so the usual Chinese Remainder Theorem doesn't apply. In fact, since they have a common factor of 4, that means that the system only has a solution when (x mod 16) and (x mod 12) are congruent mod 4. As you can see, this is only true in one of the systems listed above.

In the case where there is a solution, the minimal distance between solutions is lcm(16,12)=48 rather than 16*12 = 192 because of the common factor in the moduli. The CRT ensures that knowing x mod 12 is equivalent to knowing x mod 3 and x mod 4 since 12=3*4 and 3 and 4 are coprime. However, you already know x mod 4 because you know x mod 16 and 4 divides 16. So you can think of the extra information provided by the second equation as only being x mod 3 rather than x mod 12.

I hope this helps. I'm not sure how to explain it better without just giving the answer away.

Antimony
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