Ring Homomorphism

Ring Homomorphism
Math Olympiad uses Elementary Math up to upper-secondary school level, for Number Theory questions students are forced to memorize complex tricks without knowing the essence of Math.

As the great 19th century German Math educator Felix Klein said we should look at Elementary Math in secondary schools from the angle of higher Math, below is an example applying Abstract Algebra of "Ring Homomorphism":

Let integer N= a1a2...aj

Prove: N is divisible by 9 if
9 | (a1+ a2+ a3+...+ aj)

Proof
Let Φ: Z -> Z/9Z = {0,1,2,3...8}
Φ(z) = z mod 9

Φ is a Ring homomorphism from Ring Z to its Subring Z/9Z.

Note: Homomorphism in Math Structure (Group, Ring, etc) is like "Similarity" of triangles in Geometry.

WLOG (Without Loss of Generality)
Let j=4
N = a1a2a3a4
= a1.10³ + a2.10² + a3.10+ a4

Since in Z/9Z
Φ(aj) = aj for all j
Φ (10) = 1
Φ (10ⁿ) = Φ(10)ⁿ= 1ⁿ = 1

Φ(N)
= Φ(a1.10³ + a2.10² + a3.10+ a4)
= Φ(a1.10³ ) + Φ(a2.10² )
+ Φ(a3.10 ) + Φ(a4 )
= Φ(a1).Φ(10³ ) + Φ(a2).Φ(10² ) +
Φ(a3).Φ(10 ) + Φ(a4 )
= (a1).1 +(a2).1+ (a3).1 + (a4)
= a1+ a2+ a3+ a4

Hence,
Φ(a1a2a3a4) = a1+ a2+ a3+ a4
=> a1a2a3a4 is divisible by 9 if
9 | ( a1+ a2+ a3+ a4)

[QED]