Mathematical Rules of Divisibility

The abil­i­ty to de­ter­mine if a number is di­vis­i­ble by $3$, $6$, or $9$ is rooted in simple, el­e­gant rules.

Di­vis­i­bil­i­ty by $3$

A number is di­vis­i­ble by $3$ if and only if the sum of its digits is di­vis­i­ble by $3$.

Let $N$ be a pos­i­tive in­te­ger with digits $d_1, d_2, ..., d_n$:

Cal­cu­late the digit sum $S$:

If $S$ is di­vis­i­ble by $3$, then $N$ is di­vis­i­ble by $3$.

Ex­am­ple:
For $N = 123$,

Since $6$ is di­vis­i­ble by $3$, $123$ is di­vis­i­ble by $3$.

Di­vis­i­bil­i­ty by $9$

A number is di­vis­i­ble by $9$ if and only if the sum of its digits is di­vis­i­ble by $9$.

Ex­am­ple:
For $N = 729$,

Since $18$ is di­vis­i­ble by $9$, $729$ is di­vis­i­ble by $9$.

Di­vis­i­bil­i­ty by $6$

A number is di­vis­i­ble by $6$ if and only if it is di­vis­i­ble by both $2$ and $3$.

• To be di­vis­i­ble by $2$, the last digit of the number must be even ($0, 2, 4, 6, 8$).
• To be di­vis­i­ble by $3$, use the rule for $3$ above.

Ex­am­ple:
For $N = 132$:

• The last digit is $2$ (even), so $132$ is di­vis­i­ble by $2$.
• $1 + 3 + 2 = 6$, and $6$ is di­vis­i­ble by $3$.

There­fore, $132$ is di­vis­i­ble by $6$.