Question

1. Let a,b,c be integers,and a ≠0.if a/b, prove that a/(bx+cy),where x,y are any integers.

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Answer 1

Answer:

Definition. If a and b are integers, then a divides b if an = b for some integer n. In this case, a is a factor

or a divisor of b.

The notation a | b means “a divides b”.

The notation a 6 | b means a does not divide b

Step-by-step explanation:

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Answer 2

Answer:

1 Suppose a | b and b | c. Now a | b means that am = b for some m and b | c means that bn = c

for some n. Hence, an = c, so a | c.

2 Suppose a | x and a | y. Now a | x means that am = x for some m and a | y means that an = y for some n.

Then, bx + cy = bam + can = a(bm + cn), so a | bx + cy.

An expression of the form bx + cy is called a linear combination of x and y.

Here are some special cases of part (b):

1. If a divides x and y, then a divides x + y and x − y.

2. If a divides x, then a divides box for all b.

In the first case, apply (b) with b = 1 and c = 1 (and b = 1 and c = −1). In the second case, apply (b)

c = 0.