Which of the following numbers is divisible by 12?
53412
Checking for Divisibility by 12
To determine if a number is divisible by 12, we can use the divisibility rules. A number is divisible by 12 if and only if it is divisible by both 3 and 4.
Let's break down the rules:
- Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
- Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
Now, let's apply these rules to each of the given options.
Analyzing Option 1: 53412
- Divisibility by 4: The last two digits form the number 12. Is 12 divisible by 4? Yes, $12 \div 4 = 3$. So, 53412 is divisible by 4.
- Divisibility by 3: The sum of the digits is $5 + 3 + 4 + 1 + 2 = 15$. Is 15 divisible by 3? Yes, $15 \div 3 = 5$. So, 53412 is divisible by 3.
Since 53412 is divisible by both 3 and 4, it is divisible by 12.
Analyzing Option 2: 43412
- Divisibility by 4: The last two digits form the number 12. Is 12 divisible by 4? Yes, $12 \div 4 = 3$. So, 43412 is divisible by 4.
- Divisibility by 3: The sum of the digits is $4 + 3 + 4 + 1 + 2 = 14$. Is 14 divisible by 3? No. So, 43412 is not divisible by 3.
Since 43412 is not divisible by 3, it is not divisible by 12.
Analyzing Option 3: 33412
- Divisibility by 4: The last two digits form the number 12. Is 12 divisible by 4? Yes, $12 \div 4 = 3$. So, 33412 is divisible by 4.
- Divisibility by 3: The sum of the digits is $3 + 3 + 4 + 1 + 2 = 13$. Is 13 divisible by 3? No. So, 33412 is not divisible by 3.
Since 33412 is not divisible by 3, it is not divisible by 12.
Analyzing Option 4: 63412
- Divisibility by 4: The last two digits form the number 12. Is 12 divisible by 4? Yes, $12 \div 4 = 3$. So, 63412 is divisible by 4.
- Divisibility by 3: The sum of the digits is $6 + 3 + 4 + 1 + 2 = 16$. Is 16 divisible by 3? No. So, 63412 is not divisible by 3.
Since 63412 is not divisible by 3, it is not divisible by 12.
Based on the analysis, only 53412 is divisible by both 3 and 4, making it divisible by 12.
Summary of Divisibility Checks
| Number | Sum of Digits | Divisible by 3? | Last Two Digits | Divisible by 4? | Divisible by 12? |
|---|---|---|---|---|---|
| 53412 | 15 | Yes | 12 | Yes | Yes |
| 43412 | 14 | No | 12 | Yes | No |
| 33412 | 13 | No | 12 | Yes | No |
| 63412 | 16 | No | 12 | Yes | No |
Revision Table: Divisibility Rules
| Divisible by | Rule |
|---|---|
| 2 | The last digit is an even number (0, 2, 4, 6, or 8). |
| 3 | The sum of the digits is divisible by 3. |
| 4 | The number formed by the last two digits is divisible by 4. |
| 5 | The last digit is 0 or 5. |
| 6 | The number is divisible by both 2 and 3. |
| 9 | The sum of the digits is divisible by 9. |
| 10 | The last digit is 0. |
| 12 | The number is divisible by both 3 and 4. |
Additional Information on Number Properties
Divisibility rules are shortcuts that help us quickly determine if a number can be divided evenly by another number without performing long division. These rules are derived from the properties of numbers and place value.
When a number is divisible by two numbers that are coprime (meaning their greatest common divisor is 1), it is also divisible by their product. For example, 3 and 4 are coprime, and their product is 12. Thus, a number divisible by both 3 and 4 is also divisible by 12.
Understanding divisibility rules is useful for simplifying fractions, finding factors, and solving various problems in number theory and arithmetic.
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select the correct answer using the code given below:
