Understanding Divisibility by 12

To determine if a number is divisible by 12, we need to check if it meets two conditions simultaneously. A number is divisible by 12 if and only if it is divisible by both 3 and 4.

Divisibility Rules Explained

Let's quickly review the divisibility rules for 3 and 4:

• Divisibility Rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility Rule for 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

For a number to be divisible by 12, it must satisfy both of these rules.

Analyzing the Options for Divisibility by 12

We will now test each given number against the divisibility rules for 3 and 4.

Conclusion: Identifying the Number Divisible by 12

Based on our analysis, the number 83412 is divisible by 3 because the sum of its digits (18) is divisible by 3. It is also divisible by 4 because the number formed by its last two digits (12) is divisible by 4.

Since 83412 satisfies both the divisibility rule for 3 and the divisibility rule for 4, it is divisible by 12.

Revision Table: Common Divisibility Rules

Additional Information: Why Divisibility by 3 and 4 Means Divisibility by 12

The rule that a number is divisible by 12 if it's divisible by both 3 and 4 works because 3 and 4 are coprime numbers. Coprime numbers (or relatively prime numbers) are numbers that have no common positive divisors other than 1. Since the greatest common divisor (GCD) of 3 and 4 is 1, their product (3 × 4 = 12) inherits the divisibility properties. If a number \(N\) can be written as \(N = 3k\) (divisible by 3) and \(N = 4m\) (divisible by 4) for some integers \(k\) and \(m\), then \(N\) must contain factors of both 3 and 4. Because 3 and 4 share no common factors other than 1, the number \(N\) must essentially be a multiple of their product, 12. So, \(N\) must be expressible as \(N = 12j\) for some integer \(j\), proving its divisibility by 12.