b) Determine whether or not the statement “for all integers a, b, c, if a | bc, then a | b or a | c” is false. Justify your answer.
false
6 | 36
but 6 does ∤ either 9 or 4.
To determine whether the statement "for all integers a, b, c, if a | bc, then a | b or a | c" is false, we can provide a counterexample that disproves the statement.
Counterexample:
Let a = 2, b = 3, and c = 4.
In this case, a | bc, since 2 divides 3 * 4.
However, a does not divide b (2 does not divide 3) and it also does not divide c (2 does not divide 4).
Therefore, the statement "for all integers a, b, c, if a | bc, then a | b or a | c" is false.
Justification:
By providing a counterexample, we have shown that there exists a specific case where a | bc, but a does not divide b or c. This disproves the universal claim made by the statement and shows that it is false.
To determine whether the statement "for all integers a, b, c, if a | bc, then a | b or a | c" is false, we need to show a counterexample. A counterexample is a specific case that disproves the statement.
In this case, we can show that the statement is false by providing a specific set of integers, a, b, and c, for which a does not divide b or c, but a does divide their product, bc.
Let's consider the example where a = 3, b = 4, and c = 5.
Here, we have a case where a does not divide b (3 does not divide 4) and a does not divide c (3 does not divide 5), yet a divides their product (3 divides 4 * 5 = 20).
Since we have found a specific example where a does not divide b or c, but a divides bc, the statement "for all integers a, b, c, if a | bc, then a | b or a | c" is false.
In summary, we have disproven the statement by providing a counterexample where a does not divide b or c, but a divides bc.