Suppose p is prime and, a and b are natural numbers, and . Then either or .

Question

Suppose p is prime and, a and b are natural numbers, and  . Then either  or .

BlackTom AI · Accepted Answer

The problem states: Suppose p is prime and a and b are natural numbers, and p | a + b. Then either p | a or p | b.

To analyze this, we test the logical claim by looking for a counterexample where p divides a + b but p divides neither a nor b.

Consider a concrete prime, such as p = 3, and choose a = 1 and b = 2. Then a + b = 3, which is divisible by 3, so p | (a + b) holds.

However, 3 does not divide a = 1, and 3 does not divide b = 2 either. Thus p | a and p | b are both false in this example.

This counterexample shows that the implication 'p | a + b implies p | a or p | b' is not valid in general.

Therefore, the statement as written is not universally true. It fails for the chosen values of p, a, and b, demonstrating that divisibility of a sum by a prime does not force divisibility of either addend by that prime.

In general, this is a common confusion arising from confusion with valuations or properties specific to certain rings, but over the integers this implication does not hold for primes with arbitrary a and b.

MAT246H1 F LEC0101 Week 3 Quiz

Suppose p is prime and, a and b are natural numbers, and . Then either or .

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