Use the method of direct proof to prove the following statem

Use the method of direct proof to prove the following statements.

Solution

we know that the defination

suppose a and b are integers we say that a divides b , written a | b , if b= ac for some c in Z .in this case

we also say that a is divisor of b and that b is a multiple of a

suppose a,b,c in Z

we know that a | b means there is an integer d with b = ad ------------------------ 1

simillerly a | c means there is an integer e with c = ae -----------------------------2

now for,

b+c = ad + ae ---------------------------------------------------- (from 1 and 2)

= a(d+e)

= af --------------- ( d+e = f because d and e are integer)

by defination

that is a | b+c

hence proved