How to prove D70 = {1, 2, 5, 7, 10, 14, 35, 70} is a Boolean algebra

Question

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

1. + & . are commutative.
2. + is distributive over . and vice versa.
3. $\exists$ two distinct identity element 0, I for ($D_{70}$, +) and ($D_{70}$, .) respectively.
4. $\forall a \in D_{70}$, 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.

Answer 1

Let $x=\prod_{i=1}^{k}p_{i}^{r_{i}}$, $y=\prod_{i=1}^{k}p_{i}^{s_{i}}$ and $z=\prod_{i=1}^{k}p_{i}^{t_{i}}$ for primes $p_{i}$ and nonnegative integers $r_{i},s_{i},t_{i}$ for $i=1,\dots,k$. Then:

$$x.(y+z)=\gcd\left(x,\text{lcm}\left(y,z\right)\right)=\prod_{i=1}^{k}p_{i}^{\min\left(r_{i},\max\left(s_{i},t_{i}\right)\right)}$$

$$x.y+x.z=\text{lcm}\left(\gcd\left(x,y\right),\gcd\left(x,z\right)\right)=\prod_{i=1}^{k}p_{i}^{\max\left(\min\left(r_{i},s_{i}\right),\min\left(r_{i},t_{i}\right)\right)}$$

It remains to prove that in general: $$\min\left(r,\max\left(s,t\right)\right)=\max\left(\min\left(r,s\right),\min\left(r,t\right)\right)$$ Can you do that?

Answer 2

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.