How many of the following numbers are divisible by 3 but NOT by 9? 5826, 5964, 6039, 6336, 6489, 6564, 6867 and 6960
4
Understanding Divisibility by 3 and 9
To solve this problem, we need to understand the divisibility rules for 3 and 9.
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- A number is divisible by 9 if the sum of its digits is divisible by 9.
A key point is that any number divisible by 9 is also divisible by 3, because 9 is a multiple of 3. We are looking for numbers that are divisible by 3 BUT NOT by 9. This means we need to find numbers where the sum of the digits is divisible by 3, but the sum of the digits is not divisible by 9.
Checking Each Number
Let's check each number in the given list: 5826, 5964, 6039, 6336, 6489, 6564, 6867, and 6960.
| Number | Sum of Digits | Divisible by 3? (Sum / 3) | Divisible by 9? (Sum / 9) | Divisible by 3 but not by 9? |
|---|---|---|---|---|
| 5826 | \(5+8+2+6 = 21\) | Yes (\(21 \div 3 = 7\)) | No (\(21 \div 9\) is not an integer) | Yes |
| 5964 | \(5+9+6+4 = 24\) | Yes (\(24 \div 3 = 8\)) | No (\(24 \div 9\) is not an integer) | Yes |
| 6039 | \(6+0+3+9 = 18\) | Yes (\(18 \div 3 = 6\)) | Yes (\(18 \div 9 = 2\)) | No |
| 6336 | \(6+3+3+6 = 18\) | Yes (\(18 \div 3 = 6\)) | Yes (\(18 \div 9 = 2\)) | No |
| 6489 | \(6+4+8+9 = 27\) | Yes (\(27 \div 3 = 9\)) | Yes (\(27 \div 9 = 3\)) | No |
| 6564 | \(6+5+6+4 = 21\) | Yes (\(21 \div 3 = 7\)) | No (\(21 \div 9\) is not an integer) | Yes |
| 6867 | \(6+8+6+7 = 27\) | Yes (\(27 \div 3 = 9\)) | Yes (\(27 \div 9 = 3\)) | No |
| 6960 | \(6+9+6+0 = 21\) | Yes (\(21 \div 3 = 7\)) | No (\(21 \div 9\) is not an integer) | Yes |
Based on the checks above, the numbers that are divisible by 3 but NOT by 9 are those marked "Yes" in the last column.
- 5826
- 5964
- 6564
- 6960
There are 4 such numbers in the list.
Revision Table: Divisibility Rules in Math
Here is a quick summary of the common divisibility rules:
| Divisible by | Rule | Example |
|---|---|---|
| 2 | The last digit is even (0, 2, 4, 6, or 8). | 458 (ends in 8) |
| 3 | The sum of the digits is divisible by 3. | 123 (1+2+3=6, 6 is divisible by 3) |
| 4 | The last two digits form a number divisible by 4. | 1736 (36 is divisible by 4) |
| 5 | The last digit is 0 or 5. | 250 (ends in 0) |
| 6 | The number is divisible by both 2 and 3. | 48 (even, 4+8=12, 12 is divisible by 3) |
| 9 | The sum of the digits is divisible by 9. | 549 (5+4+9=18, 18 is divisible by 9) |
| 10 | The last digit is 0. | 780 (ends in 0) |
Additional Information on Number Properties
Understanding divisibility rules is a fundamental part of number theory and arithmetic. These rules help us quickly determine if one number can be evenly divided by another without performing long division.
- Divisibility rules are useful for simplifying fractions, finding factors, and understanding the structure of numbers.
- For instance, knowing the divisibility rule for 3 helps in identifying multiples of 3 quickly.
- Similarly, the rule for 9 is a specific case related to the sum of digits property that applies to numbers based on their base-10 representation. Any number modulo 9 is congruent to the sum of its digits modulo 9.
- These simple rules can make working with numbers much faster and easier, especially in tests or quizzes where time is limited.
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