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The least possible number with which 270! should be multiplied so that it can be exactly divisible by (54)48 is?
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The least possible number with which 270! should be multiplied so that...
Calculating the Least Possible Number for Divisibility
To find the least possible number with which 270! should be multiplied so that it can be exactly divisible by (54)48, we need to consider the highest power of each prime factor in the divisor.
Prime Factorization of (54)48
- (54)48 = 2^48 * 3^48
Prime Factorization of 270!
- 270! = 2^x * 3^y * ...
Determining the Least Possible Number
In order for 270! to be divisible by (54)48, we need to ensure that the power of 2 in the factorization of 270! is at least 48, and the power of 3 is at least 48.
Let's find the powers of 2 and 3 in 270!:
- Power of 2 in 270! = floor(270/2) + floor(270/4) + floor(270/8) + ... = 135 + 67 + 33 + ... = 135 + 67 + 33 + 16 + 8 + 4 + 2 + 1 = 266
- Power of 3 in 270! = floor(270/3) + floor(270/9) + floor(270/27) = 90 + 30 + 10 = 130
Since the powers of both 2 and 3 in 270! are already greater than 48, we don't need to multiply 270! by any additional number to make it divisible by (54)48.
Conclusion
The least possible number with which 270! should be multiplied so that it can be exactly divisible by (54)48 is 1.
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The least possible number with which 270! should be multiplied so that it can be exactly divisible by (54)48 is?
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The least possible number with which 270! should be multiplied so that it can be exactly divisible by (54)48 is? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The least possible number with which 270! should be multiplied so that it can be exactly divisible by (54)48 is? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The least possible number with which 270! should be multiplied so that it can be exactly divisible by (54)48 is?.
The least possible number with which 270! should be multiplied so that it can be exactly divisible by (54)48 is? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The least possible number with which 270! should be multiplied so that it can be exactly divisible by (54)48 is? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The least possible number with which 270! should be multiplied so that it can be exactly divisible by (54)48 is?.
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