Which of the following numbers Is divisible by 24?
49512
Finding Numbers Divisible by 24
To determine if a number is divisible by 24, we need to check if it is divisible by both 3 and 8. This is because 24 is the product of two coprime numbers, 3 and 8 ($24 = 3 \times 8$). A number is divisible by 24 if and only if it satisfies the divisibility rules for both 3 and 8.
- Divisibility Rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
- Divisibility Rule for 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
Analyzing Each Option for Divisibility by 24
Let's apply these rules to each given option:
Option 1: 52688
- Divisibility by 3: Sum of digits = $5 + 2 + 6 + 8 + 8 = 29$. 29 is not divisible by 3 ($29 \div 3 \approx 9.67$).
- Divisibility by 8: The last three digits form the number 688. $688 \div 8 = 86$. 688 is divisible by 8.
Since 52688 is not divisible by 3, it is not divisible by 24.
Option 2: 49512
- Divisibility by 3: Sum of digits = $4 + 9 + 5 + 1 + 2 = 21$. 21 is divisible by 3 ($21 \div 3 = 7$).
- Divisibility by 8: The last three digits form the number 512. $512 \div 8 = 64$. 512 is divisible by 8.
Since 49512 is divisible by both 3 and 8, it is divisible by 24.
Option 3: 64760
- Divisibility by 3: Sum of digits = $6 + 4 + 7 + 6 + 0 = 23$. 23 is not divisible by 3 ($23 \div 3 \approx 7.67$).
- Divisibility by 8: The last three digits form the number 760. $760 \div 8 = 95$. 760 is divisible by 8.
Since 64760 is not divisible by 3, it is not divisible by 24.
Option 4: 26968
- Divisibility by 3: Sum of digits = $2 + 6 + 9 + 6 + 8 = 31$. 31 is not divisible by 3 ($31 \div 3 \approx 10.33$).
- Divisibility by 8: The last three digits form the number 968. $968 \div 8 = 121$. 968 is divisible by 8.
Since 26968 is not divisible by 3, it is not divisible by 24.
Summary of Divisibility Checks
| Number | Sum of Digits | Divisible by 3? | Last 3 Digits | Divisible by 8? | Divisible by 24? |
|---|---|---|---|---|---|
| 52688 | 29 | No | 688 | Yes | No |
| 49512 | 21 | Yes | 512 | Yes | Yes |
| 64760 | 23 | No | 760 | Yes | No |
| 26968 | 31 | No | 968 | Yes | No |
Based on the analysis, only the number 49512 is divisible by both 3 and 8, making it divisible by 24.
Conclusion
The number that is divisible by 24 among the given options is 49512.
Revision Table: Understanding Divisibility Rules
| Divisor | Rule | Example (Number: 120) |
|---|---|---|
| 2 | Ends in 0, 2, 4, 6, or 8 | 120 ends in 0. Yes. |
| 3 | Sum of digits is divisible by 3 | $1+2+0 = 3$. 3 is divisible by 3. Yes. |
| 4 | Last two digits form a number divisible by 4 | Last two digits form 20. 20 is divisible by 4. Yes. |
| 5 | Ends in 0 or 5 | 120 ends in 0. Yes. |
| 6 | Divisible by both 2 and 3 | 120 is divisible by 2 and 3. Yes. |
| 8 | Last three digits form a number divisible by 8 | Last three digits form 120. $120 \div 8 = 15$. Yes. |
| 9 | Sum of digits is divisible by 9 | $1+2+0 = 3$. 3 is not divisible by 9. No. |
| 10 | Ends in 0 | 120 ends in 0. Yes. |
| 12 | Divisible by both 3 and 4 | 120 is divisible by 3 and 4. Yes. |
Additional Information: Divisibility by Composite Numbers
To check divisibility by a composite number (a number with factors other than 1 and itself), you can often break it down into its coprime factors. If a number is divisible by all its coprime factors, it is divisible by the composite number. For example:
- Divisibility by 6: Check divisibility by 2 and 3 ($6 = 2 \times 3$).
- Divisibility by 10: Check divisibility by 2 and 5 ($10 = 2 \times 5$).
- Divisibility by 12: Check divisibility by 3 and 4 ($12 = 3 \times 4$).
- Divisibility by 15: Check divisibility by 3 and 5 ($15 = 3 \times 5$).
- Divisibility by 18: Check divisibility by 2 and 9 ($18 = 2 \times 9$).
This method works because the factors used are coprime (their greatest common divisor is 1). For example, you cannot check divisibility by 4 by checking divisibility by 2 and 2, because 2 and 2 are not coprime.
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