Understanding Divisibility

Divisibility refers to the ability of one integer to be divided by another integer without leaving a remainder. When we say "a is divisible by b," we mean that a ÷ b results in a whole number.

Divisibility Rules

Divisibility rules are shortcuts that help determine if a number is divisible by another without performing the actual division:

Rule for 2

A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). For example, 246 is divisible by 2 because 6 is even.

Rule for 3

A number is divisible by 3 if the sum of its digits is divisible by 3. For example, 123 → 1+2+3 = 6, and 6 is divisible by 3.

Rule for 4

A number is divisible by 4 if its last two digits form a number that is divisible by 4. For example, 316 → 16 is divisible by 4.

Rule for 5

A number is divisible by 5 if its last digit is 0 or 5. For example, 125 ends in 5, so it is divisible by 5.

Rule for 6

A number is divisible by 6 if it is divisible by both 2 AND 3. For example, 24 is even (divisible by 2) and 2+4=6 is divisible by 3.

Rule for 9

A number is divisible by 9 if the sum of its digits is divisible by 9. For example, 729 → 7+2+9 = 18 → 1+8 = 9.

Rule for 10

A number is divisible by 10 if it ends in 0. For example, 150 ends in 0, so it is divisible by 10.

Applications

• Simplifying Fractions: Finding common divisors helps reduce fractions to their simplest form.
• Prime Testing: A number is prime if its only divisors are 1 and itself.
• Cryptography: Divisibility properties are fundamental to many encryption algorithms.
• Computer Science: Used in algorithms, data structures, and hash functions.