Draft:Factor-Digit Divisibility Criterion

Factor-Digit Divisibility Criterion (FDDC), also referred to as the Raines Factor-Digit Divisibility Theorem, is a principle in number theory concerning the divisibility of numbers in positional numeral systems with composite bases. It states that for any composite base and any nontrivial factor of that base, the divisibility of a number by that factor depends solely on the number’s least significant digit in that base.

While the underlying idea is a direct consequence of standard modular arithmetic, it has not commonly been presented as a standalone concept. The criterion was articulated by Peter Raines, who observed that the familiar "last-digit" divisibility rules for 2 and 5 in base 10 are special cases of a more general pattern applicable to any composite base.

Statement

Let $b$ be a composite integer (the base) and $f$ be a nontrivial factor of $b$ (that is, $1<f<b$). Consider a number $n$ expressed in base $b$: $n=d_{k}b^{k}+d_{k-1}b^{k-1}+\cdots +d_{1}b+d_{0}$, where $d_{0}$ is the least significant digit.

The Factor-Digit Divisibility Criterion states:

$n$ is divisible by $f$ if and only if $d_{0}$ is divisible by $f$.

Proof

Since $f\mid b$, it follows that $b\equiv 0{\pmod {f}}$, and therefore $b^{k}\equiv 0{\pmod {f}}$ for all $k\geq 1$. Substituting into the expansion of $n$: $n\equiv d_{0}{\pmod {f}}$.

This implies that $n$ is divisible by $f$ if and only if $d_{0}$ is divisible by $f$.

Examples

• Base 10:

 Factors: 2, 5  
 - Divisible by 2 if the last digit is 0, 2, 4, 6, or 8.  
 - Divisible by 5 if the last digit is 0 or 5.

• Base 15 (15 = 3 × 5):

 Label digits 0–9, A=10, B=11, C=12, D=13, E=14.  
 - Divisible by 3 if last digit is 0, 3, 6, 9, or C (12).  
 - Divisible by 5 if last digit is 0, 5, or A (10).

• Base 12 (12 = 2² × 3):

 Digits 0–9, A=10, B=11.  
 - Divisible by 2 if last digit is 0, 2, 4, 6, 8, or A.  
 - Divisible by 3 if last digit is 0, 3, 6, or 9.

Significance

The Factor-Digit Divisibility Criterion reveals that well-known divisibility tests in base 10 are not unique to the decimal system. Instead, they reflect a general property of composite bases. While the principle is straightforward from a modular arithmetic standpoint, recognizing and naming it as a standalone concept may aid in teaching and understanding divisibility rules in a broader context.

See also

• Divisibility rule
• Modular arithmetic
• Base (mathematics)
• Positional notation

Further reading

• LeVeque, William J. (1996). Fundamentals of Number Theory. Dover. ISBN 978-0486689067.
• Hardy, G. H.; Wright, E. M. (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press. ISBN 978-0199219865.
• Apostol, Tom M. (1976). Introduction to Analytic Number Theory. Springer. ISBN 978-0387901633.
• Ireland, Kenneth; Rosen, Michael (1990). A Classical Introduction to Modern Number Theory. Springer. ISBN 978-0387973296.