Understanding Divisibility by 6

To determine if a number is divisible by 6, we need to check if it meets two conditions:

• The number must be divisible by 2.
• The number must be divisible by 3.

Let's break down these rules:

1. Divisibility by 2: A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8). In other words, the number must be even.
2. Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.

A number is divisible by 6 only if both of these conditions are met.

Analyzing the Options for Divisibility by 6

Let's examine each of the given numbers to see if they are divisible by both 2 and 3.

Option 1: 34672

• Divisibility by 2: The last digit is 2, which is an even number. So, 34672 is divisible by 2.
• Divisibility by 3: Let's find the sum of the digits: \(3 + 4 + 6 + 7 + 2 = 22\). Is 22 divisible by 3? No, \(22 \div 3\) gives a remainder.

Since 34672 is divisible by 2 but not by 3, it is not divisible by 6.

Option 2: 45678

• Divisibility by 2: The last digit is 8, which is an even number. So, 45678 is divisible by 2.
• Divisibility by 3: Let's find the sum of the digits: \(4 + 5 + 6 + 7 + 8 = 30\). Is 30 divisible by 3? Yes, \(30 \div 3 = 10\).

Since 45678 is divisible by both 2 and 3, it is divisible by 6.

Option 3: 23456

• Divisibility by 2: The last digit is 6, which is an even number. So, 23456 is divisible by 2.
• Divisibility by 3: Let's find the sum of the digits: \(2 + 3 + 4 + 5 + 6 = 20\). Is 20 divisible by 3? No, \(20 \div 3\) gives a remainder.

Since 23456 is divisible by 2 but not by 3, it is not divisible by 6.

Option 4: 56792

• Divisibility by 2: The last digit is 2, which is an even number. So, 56792 is divisible by 2.
• Divisibility by 3: Let's find the sum of the digits: \(5 + 6 + 7 + 9 + 2 = 29\). Is 29 divisible by 3? No, \(29 \div 3\) gives a remainder.

Since 56792 is divisible by 2 but not by 3, it is not divisible by 6.

Summary of Divisibility Checks

Based on our analysis, only the number 45678 is divisible by both 2 and 3, making it divisible by 6.

Revision Table: Divisibility Rules

Additional Information on Divisibility Rules

Divisibility rules are helpful shortcuts for determining if a number is divisible by another number without performing the actual division. They are based on the properties of numbers and place value.

• Understanding prime factorization helps in creating divisibility rules for composite numbers like 6. Since \(6 = 2 \times 3\), a number is divisible by 6 if and only if it is divisible by its prime factors, 2 and 3.
• These rules are useful in simplifying fractions, finding factors, and solving various arithmetic problems.
• Other common divisibility rules exist for numbers like 4, 5, 9, 10, and 11. For example, a number is divisible by 4 if its last two digits form a number divisible by 4. A number is divisible by 5 if it ends in 0 or 5. A number is divisible by 9 if the sum of its digits is divisible by 9.