Which of the following numbers is divisible by 99?
60687
Understanding Divisibility by 99
To determine if a number is divisible by 99, we need to check if it is divisible by both 9 and 11, since $99 = 9 \times 11$.
Let's recall the divisibility rules for 9 and 11:
- Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
- Divisibility by 11: A number is divisible by 11 if the alternating sum of its digits (starting from the rightmost digit, alternating between adding and subtracting) is divisible by 11 (this includes 0).
Checking Each Option for Divisibility by 99
Let's apply these rules to each given option:
Option 1: 31548
- Divisibility by 9: Sum of digits = $3+1+5+4+8 = 21$. Since 21 is not divisible by 9, the number 31548 is not divisible by 9.
- Conclusion: 31548 is not divisible by 99.
Option 2: 60687
- Divisibility by 9: Sum of digits = $6+0+6+8+7 = 27$. Since 27 is divisible by 9, the number 60687 is divisible by 9.
- Divisibility by 11: Alternating sum of digits (starting from the right): $7 - 8 + 6 - 0 + 6 = 11$. Since 11 is divisible by 11, the number 60687 is divisible by 11.
- Conclusion: Since 60687 is divisible by both 9 and 11, it is divisible by 99.
Option 3: 44775
- Divisibility by 9: Sum of digits = $4+4+7+7+5 = 27$. Since 27 is divisible by 9, the number 44775 is divisible by 9.
- Divisibility by 11: Alternating sum of digits (starting from the right): $5 - 7 + 7 - 4 + 4 = 5$. Since 5 is not divisible by 11, the number 44775 is not divisible by 11.
- Conclusion: 44775 is not divisible by 99.
Option 4: 84456
- Divisibility by 9: Sum of digits = $8+4+4+5+6 = 27$. Since 27 is divisible by 9, the number 84456 is divisible by 9.
- Divisibility by 11: Alternating sum of digits (starting from the right): $6 - 5 + 4 - 4 + 8 = 9$. Since 9 is not divisible by 11, the number 84456 is not divisible by 11.
- Conclusion: 84456 is not divisible by 99.
Summary of Divisibility Checks
| Number | Sum of Digits | Divisible by 9? | Alternating Sum of Digits | Divisible by 11? | Divisible by 99? |
|---|---|---|---|---|---|
| 31548 | 21 | No | $8-4+5-1+3 = 11$ | Yes | No (Not divisible by 9) |
| 60687 | 27 | Yes | $7-8+6-0+6 = 11$ | Yes | Yes (Divisible by 9 and 11) |
| 44775 | 27 | Yes | $5-7+7-4+4 = 5$ | No | No (Not divisible by 11) |
| 84456 | 27 | Yes | $6-5+4-4+8 = 9$ | No | No (Not divisible by 11) |
Based on the checks, only the number 60687 is divisible by both 9 and 11, making it divisible by 99.
Revision Table: Key Divisibility Rules
| Divisible By | Rule | Example |
|---|---|---|
| 2 | Ends in an even digit (0, 2, 4, 6, 8). | 48 (ends in 8) |
| 3 | Sum of digits is divisible by 3. | 123 ($1+2+3=6$, 6 is div by 3) |
| 4 | The number formed by the last two digits is divisible by 4. | 516 (16 is div by 4) |
| 5 | Ends in 0 or 5. | 75 (ends in 5) |
| 6 | Divisible by both 2 and 3. | 18 (even, $1+8=9$, 9 is div by 3) |
| 9 | Sum of digits is divisible by 9. | 729 ($7+2+9=18$, 18 is div by 9) |
| 10 | Ends in 0. | 150 (ends in 0) |
| 11 | Alternating sum of digits is divisible by 11. | 1331 ($1-3+3-1=0$, 0 is div by 11) |
Additional Information on Composite Divisors
When a number's divisor is a composite number (a number with more than two factors), like 99, we can often break down the divisibility check into its prime factors or relatively prime factors. Since $99 = 9 \times 11$, and 9 and 11 are relatively prime (they share no common factors other than 1), a number is divisible by 99 if and only if it is divisible by both 9 and 11.
It is important that the factors used are relatively prime. For instance, to check divisibility by 12, we check for divisibility by 3 and 4 (since 3 and 4 are relatively prime and $3 \times 4 = 12$), not 2 and 6 (since 2 and 6 are not relatively prime).
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