Definition: Rules for determining divisibility by a number m in a base b, structurally derived from the periodic sequence of weights calculated from bⁱ mod m.
Chapter 1: The Secret Number Tricks (Elementary School Understanding)
You probably know some secret math tricks.
The "Divisible by 2" Trick: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
The "Divisible by 5" Trick: A number is divisible by 5 if its last digit is 0 or 5.
The "Divisible by 3" Trick: A number is divisible by 3 if the sum of its digits is divisible by 3. (e.g., for 141, 1+4+1=6. Since 6 is divisible by 3, 141 is too!).
These aren't magic. They are Divisibility Rules. The treatise explains that every one of these rules comes from a secret, repeating pattern of "weight" numbers.
For the "divisible by 3" rule, the secret pattern of weights is just (1, 1, 1, 1, ...)
You multiply each digit by its weight and add them up:
For 141: (1×1) + (4×1) + (1×1) = 6.
For the "divisible by 11" rule, the secret pattern is (1, -1, 1, -1, ...)
For 374: (4×1) + (7×-1) + (3×1) = 4 - 7 + 3 = 0.
The treatise shows how to find the secret pattern of weights for any number, allowing you to create your own divisibility tricks!
Chapter 2: The Weighted Sum of Digits (Middle School Understanding)
A divisibility rule is a shortcut for determining if an integer N is divisible by another integer m without performing the full division. The structural approach shows that all such rules are based on a weighted sum of the digits of N.
The key is the weight sequence. This is the repeating pattern of remainders of the powers of the base b when divided by the modulus m.
W = (b⁰ mod m, b¹ mod m, b² mod m, ...)
The Universal Rule: A number N is divisible by m if and only if the weighted sum of its digits is divisible by m.
Deriving the Rule for Divisibility by 7 in Base-10:
The Base and Modulus: b=10, m=7.
Calculate the Weight Sequence:
10⁰ = 1 ≡ 1 (mod 7)
10¹ = 10 ≡ 3 (mod 7)
10² = 100 ≡ 2 (mod 7)
10³ = 1000 ≡ 6 or -1 (mod 7)
10⁴ ≡ -3 (mod 7)
10⁵ ≡ -2 (mod 7)
10⁶ ≡ 1 (mod 7) → The cycle repeats.
The Weight Pattern (from right to left): (1, 3, 2, -1, -3, -2, ...)
The Rule: To test if a number N is divisible by 7, multiply its last digit by 1, the next by 3, the next by 2, etc., add them up, and see if the result is divisible by 7.
Example: Test N = 4851
(1×1) + (5×3) + (8×2) + (4×-1)
= 1 + 15 + 16 - 4 = 28
Since 28 is divisible by 7, the original number 4851 must also be divisible by 7. (Check: 4851 / 7 = 693).
Chapter 3: The Law of Remainder Translation (High School Understanding)
The structural derivation of all Divisibility Rules comes from a single, powerful theorem: the Law of Remainder Translation.
The Theorem: For any integer N with digits (...d₂d₁d₀)_b, its remainder modulo m is given by:
N mod m ≡ [ Σ dᵢ(bⁱ mod m) ] mod m
This law translates a property of the number's Algebraic Soul (N mod m) into a calculation on its Arithmetic Body (the sum of its digits dᵢ, each multiplied by a weight bⁱ mod m).
A number N is divisible by m if and only if N mod m = 0. Therefore, the divisibility rule is:
Σ dᵢ(bⁱ mod m) ≡ 0 (mod m)
Deriving Familiar Rules:
Divisibility by b-1 (like 9 in base-10):
The weight sequence is bⁱ mod (b-1).
Since b ≡ 1 (mod b-1), then bⁱ ≡ 1ⁱ ≡ 1 (mod b-1) for all i.
The weight sequence is (1, 1, 1, ...).
The rule becomes Σ dᵢ(1) ≡ 0 (mod b-1). This is the familiar "sum of digits" rule.
Divisibility by b+1 (like 11 in base-10):
The weight sequence is bⁱ mod (b+1).
Since b ≡ -1 (mod b+1), then bⁱ ≡ (-1)ⁱ (mod b+1).
The weight sequence is (1, -1, 1, -1, ...).
The rule becomes d₀ - d₁ + d₂ - ... ≡ 0 (mod b+1). This is the "alternating sum of digits" rule.
This structural approach shows that these "tricks" are not random; they are necessary consequences of the modular behavior of the base b.
Chapter 4: A Consequence of the Polynomial Representation (College Level)
The most formal understanding of Divisibility Rules comes from viewing a number's representation as a polynomial.
An integer N with digits (dₖ...d₀)_b is equivalent to the polynomial P(x) = Σ dᵢxⁱ evaluated at x=b.
N = P(b)
A number N is divisible by m if N ≡ 0 (mod m). This is equivalent to:
P(b) ≡ 0 (mod m)
The Law of Remainder Translation is a direct consequence of the properties of ring homomorphisms. The map φ(x) = x mod m is a homomorphism. Therefore:
P(b) mod m = P(b mod m) mod m
N mod m = (Σ dᵢ(b mod m)ⁱ) mod m
N mod m = (Σ dᵢ(bⁱ mod m)) mod m
This confirms that the divisibility rule is a statement about evaluating the number's representative polynomial at the value b mod m.
The Weight Sequence as a Group Cycle:
The weight sequence W = (b⁰, b¹, b², ...) modulo m is the sequence of elements generated by b in the multiplicative group of integers modulo m, (ℤ/mℤ)ˣ (if b and m are coprime). The length of this sequence is the order of b in that group.
The Divisibility Rules are therefore a practical application of group theory. The "weights" are the elements of the cyclic subgroup generated by the base b within the modular ring ℤ/mℤ. This provides a complete, structural, and predictable foundation for creating a divisibility test for any m in any base b.
Chapter 5: Worksheet - Creating the Tricks
Part 1: The Secret Number Tricks (Elementary Level)
The "divisible by 10" trick is that a number must end in 0. The pattern of weights for this is (1, 0, 0, 0, ...). Test this with the number 120: (0×1) + (2×0) + (1×0) = 0.
What do you think the pattern of weights is for the "divisible by 2" trick in base-10?
Part 2: The Weighted Sum (Middle School Understanding)
Let's derive the divisibility rule for 8 in base-10.
What is 10⁰ mod 8?
What is 10¹ mod 8?
What is 10² mod 8?
What is 10³ mod 8?
What is the weight sequence W for this rule?
Use your rule to test if the number N=136 is divisible by 8.
Part 3: The Law of Remainder Translation (High School Understanding)
Use the structural derivation to prove the "sum of digits" rule for divisibility by 3 in base-10.
Use the structural derivation to prove the "alternating sum of digits" rule for divisibility by 7 in base-6.
Part 4: The Group Cycle (College Level)
What is the order of the element 2 in the group (ℤ/11ℤ)ˣ? What does this tell you about the length of the weight sequence for the divisibility-by-11 test in base-2?
What is a primitive root? If a base b is a primitive root modulo a prime p, what does this imply about the length and complexity of its divisibility rule?
Explain how the Divisibility Rules are a practical application of the properties of the quotient ring ℤ/mℤ.