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

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

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.