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Which is the greatest 4-digit perfect square exactly divisible by 3, 5, 7 and 9?
- a)9,999
- b)9,684
- c)9,801
- d)No such number exists.
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Which is the greatest 4-digit perfect square exactly divisible by 3, ...
As the number has to be divisible by 3, 5, 7 and 9 (or 3 x 3), the number is of the form a x 3 x 3 x 5 x 7, where a is an integer.
As the number is a perfect square number, the number is of the form b x 3 x 3 x 5 x 5 x 7 x 7, where b is a square number as well.
But 3 x 3 x 5 x 5 x 7 x 7 = 11025 > 9999, which is the largest 4-digit number.
So, no such number exists.
Hence, answer option 4 is correct.
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Community Answer
Which is the greatest 4-digit perfect square exactly divisible by 3, ...
Understanding the Problem
To identify the greatest 4-digit perfect square divisible by 3, 5, 7, and 9, we need to first determine the least common multiple (LCM) of these numbers.
Calculating the LCM
- The prime factorization of each number is:
- 3 = 3
- 5 = 5
- 7 = 7
- 9 = 3^2
- The LCM is obtained by taking the highest power of each prime factor:
- LCM = 3^2 * 5 * 7 = 315
Finding the Greatest 4-Digit Perfect Square
- The largest 4-digit number is 9999.
- To find the largest perfect square less than or equal to 9999, we need to calculate the square root of 9999, which is approximately 99.995, indicating the largest integer is 99.
- The largest perfect square is 99^2 = 9801.
Checking Divisibility
Now, we need to check if 9801 is divisible by 315:
- Dividing 9801 by 315 gives approximately 31.116, which shows it is not divisible.
Conclusion
Since 9801 is the largest 4-digit perfect square and is not divisible by 315, we conclude:
- There is no 4-digit perfect square that meets the criteria of being divisible by 3, 5, 7, and 9.
Thus, the correct answer is option D: No such number exists.
To identify the greatest 4-digit perfect square divisible by 3, 5, 7, and 9, we need to first determine the least common multiple (LCM) of these numbers.
Calculating the LCM
- The prime factorization of each number is:
- 3 = 3
- 5 = 5
- 7 = 7
- 9 = 3^2
- The LCM is obtained by taking the highest power of each prime factor:
- LCM = 3^2 * 5 * 7 = 315
Finding the Greatest 4-Digit Perfect Square
- The largest 4-digit number is 9999.
- To find the largest perfect square less than or equal to 9999, we need to calculate the square root of 9999, which is approximately 99.995, indicating the largest integer is 99.
- The largest perfect square is 99^2 = 9801.
Checking Divisibility
Now, we need to check if 9801 is divisible by 315:
- Dividing 9801 by 315 gives approximately 31.116, which shows it is not divisible.
Conclusion
Since 9801 is the largest 4-digit perfect square and is not divisible by 315, we conclude:
- There is no 4-digit perfect square that meets the criteria of being divisible by 3, 5, 7, and 9.
Thus, the correct answer is option D: No such number exists.
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Which is the greatest 4-digit perfect square exactly divisible by 3, 5, 7 and 9?a)9,999b)9,684c)9,801d)No such number exists.Correct answer is option 'D'. Can you explain this answer?
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