Divisibility Rules for 3 and 6

Divisibility Rules

Divisibility By 3 With Sum Of Digits

• A number is divisible by 3 if the sum of its digits is also divisible by 3.
• Example: 123 is divisible by 3 because 1 + 2 + 3 = 6, which is divisible by 3.

Divisibility By 6 With Even Number And Sum Of Digits

• A number is divisible by 6 if it is even (last digit is 0, 2, 4, 6, or 8) and the sum of its digits is divisible by 3.
• Example: 120 is divisible by 6 because it is even (last digit is 0) and 1 + 2 + 0 = 3, which is divisible by 3.

Divisibility By 9 With Sum Of Digits And Remainder

• A number is divisible by 9 if the sum of its digits is divisible by 9 or if the remainder when divided by 9 is 0.
• Example: 459 is divisible by 9 because 4 + 5 + 9 = 18, which is divisible by 9.

Relationship Between Divisibility Rules

• If a number is divisible by 3 and 2, it is also divisible by 6.
• If a number is divisible by 9, it is also divisible by 3.

Applying Divisibility Rules To Real-world Problems

• Divisibility rules can be used to quickly check if a number is divisible by a certain number without having to perform the actual division.
• Examples: checking if a number is divisible by 3 or 6 when splitting a group of people into teams, or checking if a number is divisible by 9 when calculating the total cost of items.

Prime Factors

• A prime factor is a prime number that divides a given number exactly without leaving a remainder.
• Example: the prime factors of 12 are 2, 2, and 3 because 2 × 2 × 3 = 12.
• Prime factors are used to find the greatest common divisor (GCD) and least common multiple (LCM) of two or more numbers.

Divisibility Rules

Divisibility by 3

• A number is divisible by 3 if the sum of its digits is also divisible by 3.
• This rule can be applied to any number, regardless of its size.

Divisibility by 6

• A number is divisible by 6 if it is even (last digit is 0, 2, 4, 6, or 8) and the sum of its digits is divisible by 3.
• This rule combines two conditions: even number and sum of digits being divisible by 3.

Divisibility by 9

• A number is divisible by 9 if the sum of its digits is divisible by 9 or if the remainder when divided by 9 is 0.
• This rule provides two alternative ways to check for divisibility by 9.

Relationships Between Divisibility Rules

• If a number is divisible by 3 and 2, it is also divisible by 6.
• If a number is divisible by 9, it is also divisible by 3.
• These relationships can be used to simplify divisibility checks.

Applications of Divisibility Rules

• Divisibility rules can be used to quickly check if a number is divisible by a certain number without performing actual division.
• Examples include checking divisibility by 3 or 6 when splitting a group into teams, or checking divisibility by 9 when calculating the total cost of items.

Prime Factors

• A prime factor is a prime number that divides a given number exactly without leaving a remainder.
• Prime factors are used to find the greatest common divisor (GCD) and least common multiple (LCM) of two or more numbers.
• Example: the prime factors of 12 are 2, 2, and 3, as 2 × 2 × 3 = 12.