Divisibility Rules Review

Foundations of Divisibility Rules

Divisibility rules are specialized mathematical guidelines that help determine if one specific number is divisible by another without the requirement of performing a full long division calculation. These rules allow for the efficient identification of factors and divisors for large values.

Rule for Divisibility by $2$

According to the rules, if the last digit of the number is an even number or $0$, then the number is divisible by $2$.

For example, in the case of the number $62$, it is divisible by $2$ because the last digit of the number is $2$, which is an even number.

Rule for Divisibility by $3$

If the sum of all the digits in the number and can be devided by $3$. then it is divisible by $3$.

For example, consider the number $123$. This number is divisible by $3$ because the calculation $(1+2)+3=6$ results in $6$, and $6$ can be divided by $3$.

Rule for Divisibility by $4$

If the number formed by its last digits can be divided by $4$, or if it's two zeros, Then it is divisible by $4$.

For example, in the number $21,536$, the last two digits of the number are $36$. Since $36$ can be divided by $4$, the entire number $21,536$ is divisible by $4$.

Rule for Divisibility by $5$

If the last digit of the number is either $0$ or $5$, then it is divisible by $5$.

For example, the number $26,075$ is divisible by $5$ because the last digit of the number is $5$.

Rule for Divisibility by $6$

If the number is divisible by $2$ and $3$, then it is divisible by $6$.

For example, the number $123,468$ is divisible by $6$ because it fulfills both the rule for $2$ and the rule for $3$:

• For $2$: The last digit is $8$, which is an even number.
• For $3$: The sum calculation $(1+2)+(3+4)+(6+8)=24$ shows that $24$ can be divided by $3$.

Rule for Divisibility by $8$

If the number formed from its last $3$ digits can be divided by $8$, then it is divisible $8$.

For example, in the number $43,739,112$, the last three digits of the number are $112$. Because $112$ can be divided by $8$, the number is divisible by $8$.

Rule for Divisibility by $9$

If the sum of all the digits in the number and can be divided by $9$ then it is divisible by $9$.

For example, the number $92,817,234$ is divisible by $9$ because the sum of the digits $(9+2)+(8+1)+(7+2)+(3+4)=36$ and $36$ can be divided by $9$.

Rule for Divisibility by $10$

If the last digit of the number is , then it is divisible by $10$.

For example, the number $2,134,587,690$ is divisible by $10$ because the last digit is $0$.

TIP: If the number divisibe by $2$ or $5$, it is divisibe by $10$.

Rule for Divisibility by $11$

If the difference of the sums of all the digits in the odd and even positions can be divided by $11$ or is all, then it is divisible by $11$.

For example, using the number $47806$ and identifying positions as OEOE (Odd and Even):

• Odd position sum: $(8+6)+4=18$
• Even position sum: $4+8+ 67+0=7$ (given as the sum of digits in even positions)
• Difference: $18-7=11$
• Verification: You can devided $11$ by $11$. Thus, the number is divisible by $11$.

Rule for Divisibility by $12$

If the number is divisible by $4$ and $3$ then it is divisibility.

For example, the number $87,612$ is tested as follows:

• Divisibility by $3$: $(8+7)+(6+2)+1=24$ and $24$ can be divided by $3$.
• Divisibility by $4$: The last $2$ digits of the number is $24$ and $24$ can be divided by $4$.