Which of the following numbers is divisible by 36 ?
55512
Finding Numbers Divisible by 36
To determine if a number is divisible by 36, we can use divisibility rules. Since $36 = 4 \times 9$ and 4 and 9 are coprime (they have no common factors other than 1), a number is divisible by 36 if and only if it is divisible by both 4 and 9.
Divisibility Rules Explained
- Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
- Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
Checking Each Option for Divisibility by 36
Let's apply the divisibility rules for 4 and 9 to each given option:
Option 1: 8840
- Divisibility by 4: The last two digits are 40. Since $40 \div 4 = 10$, 8840 is divisible by 4.
- Divisibility by 9: The sum of the digits is $8 + 8 + 4 + 0 = 20$. Since 20 is not divisible by 9, 8840 is not divisible by 9.
- Conclusion: Since 8840 is not divisible by 9, it is not divisible by 36.
Option 2: 1542
- Divisibility by 4: The last two digits are 42. Since $42 \div 4$ leaves a remainder (e.g., $42 = 4 \times 10 + 2$), 1542 is not divisible by 4.
- Conclusion: Since 1542 is not divisible by 4, it is not divisible by 36. We don't need to check for divisibility by 9.
Option 3: 96272
- Divisibility by 4: The last two digits are 72. Since $72 \div 4 = 18$, 96272 is divisible by 4.
- Divisibility by 9: The sum of the digits is $9 + 6 + 2 + 7 + 2 = 26$. Since 26 is not divisible by 9, 96272 is not divisible by 9.
- Conclusion: Since 96272 is not divisible by 9, it is not divisible by 36.
Option 4: 55512
- Divisibility by 4: The last two digits are 12. Since $12 \div 4 = 3$, 55512 is divisible by 4.
- Divisibility by 9: The sum of the digits is $5 + 5 + 5 + 1 + 2 = 18$. Since $18 \div 9 = 2$, 55512 is divisible by 9.
- Conclusion: Since 55512 is divisible by both 4 and 9, it is divisible by 36.
Based on the analysis, only 55512 is divisible by 36.
| Number | Divisible by 4 (Check last 2 digits) | Divisible by 9 (Check sum of digits) | Divisible by 36 (Both must be Yes) |
|---|---|---|---|
| 8840 | Yes (40 is div by 4) | No (Sum=20, not div by 9) | No |
| 1542 | No (42 is not div by 4) | - | No |
| 96272 | Yes (72 is div by 4) | No (Sum=26, not div by 9) | No |
| 55512 | Yes (12 is div by 4) | Yes (Sum=18, div by 9) | Yes |
Revision Table: Divisibility Rules
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Ends in 0, 2, 4, 6, or 8 | 128 (ends in 8) |
| 3 | Sum of digits is divisible by 3 | 123 (1+2+3=6, 6 is div by 3) |
| 4 | Last two digits form a number divisible by 4 | 116 (16 is div by 4) |
| 5 | Ends in 0 or 5 | 150 (ends in 0) |
| 6 | Divisible by both 2 and 3 | 132 (ends in 2, sum=6) |
| 9 | Sum of digits is divisible by 9 | 162 (1+6+2=9, 9 is div by 9) |
| 10 | Ends in 0 | 140 (ends in 0) |
| 11 | Alternating sum of digits is divisible by 11 | 121 (1-2+1=0, 0 is div by 11) |
Additional Information: Combined Divisibility Rules
When checking divisibility by a composite number (a number with more than two factors, like 36), you can break it down into its prime factors or coprime factors. For example:
- To check divisibility by 6, check divisibility by 2 and 3 (since 2 and 3 are coprime factors of 6).
- To check divisibility by 10, check divisibility by 2 and 5 (since 2 and 5 are coprime factors of 10).
- To check divisibility by 12, check divisibility by 3 and 4 (since 3 and 4 are coprime factors of 12).
- To check divisibility by 15, check divisibility by 3 and 5 (since 3 and 5 are coprime factors of 15).
Using coprime factors is important. For example, checking divisibility by 12 cannot be done by checking divisibility by 2 and 6, because 2 and 6 are not coprime (they share a common factor of 2).
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