Which number among 34936, 35508, 35580, and 36508 is divisible by 33?
When faced with a set of numbers and asked to identify which one is divisible by another specific number, like 33 in this case, we can employ a few reliable mathematical techniques. The question at hand asks us to pinpoint which number from the list: 34936, 35508, 35580, and 36508, will divide evenly by 33, leaving no remainder.
Understanding Divisibility by 33
A number is divisible by 33 if it is divisible by both 3 and 11, as 3 and 11 are prime factors of 33 and they don't share any common factors other than 1.
Divisibility Rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
Divisibility Rule for 11
A number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at even places is either 0 or a multiple of 11.
Testing Each Number
Let's apply these rules to each number in the given list:
1. Testing 34936
- Divisibility by 3: Sum of digits = 3 + 4 + 9 + 3 + 6 = 25. Since 25 is not divisible by 3, 34936 is not divisible by 3. Therefore, it cannot be divisible by 33.
2. Testing 35508
- Divisibility by 3: Sum of digits = 3 + 5 + 5 + 0 + 8 = 21. Since 21 is divisible by 3 (21 ÷ 3 = 7), 35508 is divisible by 3.
- Divisibility by 11:
- Digits at odd places (from right): 8, 5, 3. Sum = 8 + 5 + 3 = 16.
- Digits at even places (from right): 0, 5. Sum = 0 + 5 = 5.
- Difference = 16 - 5 = 11. Since 11 is divisible by 11, 35508 is divisible by 11.
Since 35508 is divisible by both 3 and 11, it is divisible by 33.
3. Testing 35580
- Divisibility by 3: Sum of digits = 3 + 5 + 5 + 8 + 0 = 21. Since 21 is divisible by 3, 35580 is divisible by 3.
- Divisibility by 11:
- Digits at odd places (from right): 0, 5, 3. Sum = 0 + 5 + 3 = 8.
- Digits at even places (from right): 8, 5. Sum = 8 + 5 = 13.
- Difference = 13 - 8 = 5. Since 5 is not divisible by 11, 35580 is not divisible by 11. Therefore, it is not divisible by 33.
4. Testing 36508
- Divisibility by 3: Sum of digits = 3 + 6 + 5 + 0 + 8 = 22. Since 22 is not divisible by 3, 36508 is not divisible by 3. Therefore, it cannot be divisible by 33.
Conclusion
Based on our tests:
- 34936 is not divisible by 3.
- 35508 is divisible by both 3 and 11, making it divisible by 33.
- 35580 is divisible by 3 but not by 11.
- 36508 is not divisible by 3.
Therefore, the number among 34936, 35508, 35580, and 36508 that is divisible by 33 is 35508.
You can confirm this by performing the division:
35508 ÷ 33 = 1076
This results in a whole number, confirming our finding.
Frequently Asked Questions (FAQ)
How do I check if a number is divisible by 3?
To check if a number is divisible by 3, you add up all of its digits. If the sum of the digits is a number that can be divided evenly by 3 (like 3, 6, 9, 12, 15, etc.), then the original number is also divisible by 3.
Why is divisibility by 3 and 11 important for checking divisibility by 33?
The number 33 is a composite number, meaning it can be broken down into smaller factors. The prime factors of 33 are 3 and 11. If a number can be divided evenly by both 3 and 11 without any remainder, it will also be divisible by their product, which is 33.
What is the rule for checking divisibility by 11?
To check if a number is divisible by 11, you take the digits of the number and alternate adding and subtracting them, starting from the rightmost digit. For example, for a number like 12345, you would calculate 5 - 4 + 3 - 2 + 1. If the final result is 0 or any multiple of 11 (like 11, 22, -11, etc.), then the original number is divisible by 11.
Can I just divide each number by 33 to find the answer?
Yes, you can definitely perform the division directly. For example, you could calculate 34936 ÷ 33, then 35508 ÷ 33, and so on. The number that results in a whole number (no decimal or remainder) is the one divisible by 33. The divisibility rules are often used as a quicker method, especially when dealing with larger numbers or when you want to avoid lengthy calculations.
