1. `D_(10)` is a boolean algebra?

Solution:
(1) Commutative laws
The following are the tables for `+`,`*` and `'`

`+` 1 2 5 10 1  1 `1=1+1=lcm(1,1)`  2 `2=1+2=lcm(1,2)`  5 `5=1+5=lcm(1,5)`  10 `10=1+10=lcm(1,10)` 2  2 `2=2+1=lcm(2,1)`  2 `2=2+2=lcm(2,2)`  10 `10=2+5=lcm(2,5)`  10 `10=2+10=lcm(2,10)` 5  5 `5=5+1=lcm(5,1)`  10 `10=5+2=lcm(5,2)`  5 `5=5+5=lcm(5,5)`  10 `10=5+10=lcm(5,10)` 10  10 `10=10+1=lcm(10,1)`  10 `10=10+2=lcm(10,2)`  10 `10=10+5=lcm(10,5)`  10 `10=10+10=lcm(10,10)`

`*` 1 2 5 10 1  1 `1=1*1=gcd(1,1)`  1 `1=1*2=gcd(1,2)`  1 `1=1*5=gcd(1,5)`  1 `1=1*10=gcd(1,10)` 2  1 `1=2*1=gcd(2,1)`  2 `2=2*2=gcd(2,2)`  1 `1=2*5=gcd(2,5)`  2 `2=2*10=gcd(2,10)` 5  1 `1=5*1=gcd(5,1)`  1 `1=5*2=gcd(5,2)`  5 `5=5*5=gcd(5,5)`  5 `5=5*10=gcd(5,10)` 10  1 `1=10*1=gcd(10,1)`  2 `2=10*2=gcd(10,2)`  5 `5=10*5=gcd(10,5)`  10 `10=10*10=gcd(10,10)`

`'` 1 2 5 10  10 `10=1'`  5 `5=2'`  2 `2=5'`  1 `1=10'`

From the tables it is clear that `AA x, y in D_(10), x+yin D_(10) and x*yin D_(10)`

`:.` `+` and `*` are binary operations.

The symmetry about the principal diagonal of first two tables indicates the commutative laws hold.

(2) Associative laws
We verify associative laws for 1,2 and 5
`1+(2+5)=1+10=10`

`(1+2)+5=2+5=10`

and
`1*(2*5)=1*1=1`

`(1*2)*5=1*5=1`

Similarly, it may be verified for other cases.

(3) Distributive laws
We verify distributive laws for 2,5 and 10
`2+5*10=2+5=10`

`(2+5)*(2+10)=10*10=10`

and
`2*(5+10)=2*10=2`

`2*5+2*10=1+2=2`

Similarly, it may be verified for other cases.

(4) Existence of identity elements
For the zero and unit elements, we check
`1+1=1,1+2=2,1+5=5,1+10=10`

`10*1=1,10*2=2,10*5=5,10*10=10`

`:. 1` is the zero element and `10` is the unit element


(5) Existence of complement
`1+1'=1+10=10` & `1*1'=1*10=1`
`2+2'=2+5=10` & `2*2'=2*5=1`

`1'=10,``10'=1`
`2'=5,``5'=2`

`:. (D_(10),+,*,',1,10)` is a boolean algebra.