Could somebody answer part 1 and 5 Thank you A Properties of

Could somebody answer part (1) and (5)?

Thank you

A. Properties of the Relation a Divides b Prove the following, for any integers a, b, and C:

Solution

1) a divides b hence b = am for some integer m

b divides c and hence c = bn for some integer n.

Together c= bn = amn=(mn) a where mn is an integer

or a|c is true.

2) Let a divide b.

b = ma = -m(-a)

-b = -m(a)

Thus a divides -b and -a divides b.
Now assume that a divides -b and also -a divides b.

Then -b = la and b =-na

Or b =(-l)a and b = -n(-a)

Hence it follows that

a|(-b) and (-a)|b

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3) BY the second part proved we can 1|a implies (-1)|a

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4) 0 = a x0 for any a

Hence a|0

5) C|a implies a = cm, similarly c|b implies b = cn

ax+by = acm+bcn = c(am+bn)

Thus c|(ax+by)