Prove the following statement directly from the definitions of the terms. Do not use any other facts previously proved in class or in the text or in the exercises.

Question

For all integers a, b, and c, if a | b and a | c, then a | (5b + 3c).

Answer 1

Proof : Suppose a, b, and c are any integers such that a | b and a | c. [We must show that a | (5b+3c).]
By definition of divisibility, b = ar and c = as for some integers r and s. Then
5b + 3c = 5(ar) + 3(as) by substitution
= a(5r + 3s) by the commutative and associative laws of algebra.
Let t = 5r+3s. Then t is an integer because products and sums of integers are integers, and 5b+3c = at.
Thus, by definition of divisibility, a | (5b + 3c) [as was to be shown].
Prove the following statement directly from the definitions of the terms. Do not use any other facts
previously proved in class or in the text or in the exercises.
For all integers a; b; and c; if a | b and a | c; then a | (5b + 3c):