How to prove D70 = {1, 2, 5, 7, 10, 14, 35, 70} is a Boolean algebra Prove that the set $D_{70}$ = {1, 2, 5, 7, 10, 14, 35, 70} of positive factors is a Boolean algebra under the operation (+), (.), (') defined by 
$$x + y = lcm(x, y)$$
$$x . y = gcd(x, y)$$
$$x' = \frac{70}{x}$$
Attempt:
To Prove $D_{70}$ is a boolean algebra we have to satisfy below 4 properties of boolean algebra


*

*+ & . are commutative.

*+ is distributive over . and vice versa.

*$\exists$ two distinct identity element 0, I for ($D_{70}$, +) and ($D_{70}$, .) respectively.

*$\forall a \in D_{70}$, a + a' = I & a + a' = 0


Property 1: Satisfied since lcm, gcd are commutative
Property 2: ?
Property 3:  I = 70, 0 = 1
Property 4: Satisfied
I'm having trouble proving the Property 2 can anybody help.

Write out the operation tables for "+", ".", and " ' ".
Then check that for all x, y, z in D70 under those operations that the distributive laws hold.
You could write a computer program for the last part, use OTTER, or treat those tables as an 8-valued logical system and write out truth tables for the computation.
