Given integers abcd verify the following a if ab then abc b

Given integers a,b,c,d, verify the following:

a.) if a|b, then a|bc

b.) if a|b and ac, then a^2 | bc

c.) a|b if and only if ac|bc, where c=/=0

d.) if a|b and c|d, then ac|bd

please show work

Solution

a)

a|b implies :b=ma for some integer m

Hence, bc=mca ie bc is multiple of a

Hence, a|bc

b)

False

a|ac for all c because :ac is a multiple of ac for all integers c

Let, a=2,b=2

Let, c=3

Hence, bc=6 and a|ac, a|b

a^2=4

But, a^2 =4 does not divide bc

c)

Let, ac|bc

c is not equal to zero.

Hence, ac=mbc or

c(a-mb)=0

Since c is non zero . Hence, a-mb=0 or a=mb

Hence, a|b

Now let, a|b

Hence, b=ma

Hence, bc=mac

Hence, ac|bc

Hence proved

d)

a|b means ,b=ma for some integer m

c|d means ,d=nc for some integer n

Hence,

bd=ma*nc=mn*ac

Hence, ac|bd