Prove that for all positive integers a, b, c, if b is divisible by a and b is not divisible by c, then a is not divisible by c.

Step 1: Understanding the Problem We need to prove that for any positive integers a, b, c, if b is divisible by a and b is not divisible by c, then a is not divisible by c.

Step 2: Assumptions We are given that b is divisible by a, which means there exists an integer k such that b = ak. We are also given that b is not divisible by c, which means there does not exist an integer m such that b = mc.

Step 3: Proof We need to prove that a is not divisible by c. Let's assume, for the sake of contradiction, that a is divisible by c. This means there exists an integer n such that a = nc.

Step 4: Substitution If a = nc, then we can substitute a in the equation b = ak to get b = nck.

Step 5: Contradiction But this implies that b is divisible by c (since b = nck), which contradicts our assumption that b is not divisible by c. Therefore, our initial assumption that a is divisible by c must be false.

Step 6: Conclusion Therefore, if b is divisible by a and b is not divisible by c, then a is not divisible by c.