Question

If a,b, and c are integers and if a≠0, which of these must be true? a. If a∣b and a∣c, then a∣−5b+4c. b. If a∣b and a∣c. then a∣b−c. c. If a∣b and a∣c, then a−b⋅c∣a+b⋅c. d. If a∣b and a∣c, then 2a∣3b+5c e. If a∣b and a∣c, then a∣7b+3c.

Answer 1

If a∣b and a∣c, then 2a∣3b+5c.

Step-by-step explanation:

To determine which statement must be true if a∣b and a∣c, we can test each option by substituting values for a, b, and c. Let's test option d. If a∣b and a∣c, then we can say that b = ka and c = la for some integers k and l. Substituting these values into the expression 3b+5c, we get:

3b+5c = 3(ka)+5(la) = (3k+5l)a

Since (3k+5l) is an integer, we can conclude that 2a∣3b+5c. Therefore, option d must be true.