Original challenging Question proposed by MEA

Updated original

((D(35)⊕D(56)) => ¬D(a)^D(14)) v E(3) ≡ 1

Replace implication by disjunction

(D(35) ≡ D(56)) v ¬D(a)^D(2)^D(7) v E(3) ≡ 1

First notice

E(3)=E(1)+E(2) = ¬(¬E(1)^¬E(2)) = ¬D(4)

So, converting  to following below

(D(35) ≡ D(56)) v ¬D(a)^D(2)^D(7) v  ¬D(4) ≡ 1
D(35)^D(56) v ¬D(35)^¬D(56) v ¬D(A)^D(7) v ¬D(4) ≡ 1

D(35)^D(56) = D(7)^D(5)^D(8)

¬D(35)^¬D(56) = (¬D(5) v ¬D(7))^(¬D(8) v ¬D(7) =
  = ¬D(5)^¬D(8) v ¬D(7)
D(7)^D(5)^D(8) v ¬D(5)^¬D(8) v ¬D(7) v
                     v ¬D(A)^D(7) v ¬D(4) ≡ 1
D(5)^D(8) v ¬D(5)^¬D(8) v
                  v ¬D(7) v ¬D(A) v ¬D(4) ≡ 1
D(40) v ¬D(5)^¬D(8) v ¬D(28) v ¬D(A) ≡ 1

Thus A(min) = 40

Solution of challenging equation with mixed basic predicates of type E() && DEL() via technique proposed by Helen Mironchicк.

Denote by DEL (n, m) the statement "a natural number n is divided without a remainder by a positive integer m". For what is the smallest natural number A, the formula
    (D(34)⊕D(51) => ¬D(A)^D(272)) v Е(15) ≡ 1  
is identically true (that is, it takes the value 1 for any natural value of the variable x)?

First notice belongs to Helen :-

D(16) = ¬E(1)*¬E(2)*¬E(4)*¬E(8) = 
   =  ¬(E(1) v E(2) v E(4) v E(8)) = ¬E(15)
E(16)=¬D(16)

Convert original equation from implication to disjunction

(D(34)≡D(51)) v ¬D(A)^D(272)) v ¬D(16) ≡ 1
D(34)^D(51) v ¬D(34)^¬D(51) v ¬D(A)^D(272)) v ¬D(16) ≡ 1
D(34)^D(51) v ¬D(34)^¬D(51) v ¬D(A)^D(17)^D(16) v ¬D(16) ≡ 1
 
Supress D(17) in first and fourth terms via ¬D(17),
then supress D(2^4) via ¬D(2^4) in fourth term

D(34) = D(2)^D(17)
D(51) = D(3)^D(17)
D(31)*D(51) = D(2)^D(3)^D(17)

¬D(34) = (¬D(2) v ¬D(17))
¬D(51) = (¬D(3) v ¬D(17))

¬D(34)*¬D(34) = ¬D(2)*¬D(3) v ¬D(17)

D(2)^D(3)^D(17) v ¬D(2)^¬D(3) v ¬D(17) v 
  v  ¬D(A)^D(17)^D(2^4) v ¬D(2^4) ≡ 1

D(6) v ¬D(2)^¬D(3) v ¬D(17) v ¬D(A) v ¬D(2^4) ≡ 1

Thus A(min) = 6

Refences 
1. A. Mironchik, ALGEBRA OF PREDICATES AND RELATED GEOMETRIC MODELS CREATION IN REGARDS OF UNIFIED STATE
EXAMINATION IN INFORMATICS (RUSSIAN EGE) ,
Informatics at school №3 , 2019