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What is the least number by which 13720 must be divided so that the quotient is a perfect cube
- a)4
- b)2
- c)7
- d)5
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
What is the least number by which 13720 must be divided so that the qu...
Write 13720 as product of prime numbers
13720 =2*2*2*7*7*7*5
=(2*2*2)*(7*7*7)*5
Only 5 is isolated
To make 13720 perfect cube we have to divide it with 5
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What is the least number by which 13720 must be divided so that the qu...
Solution:
To find the least number by which 13720 must be divided to get a perfect cube quotient, we need to factorize the given number into its prime factors.
Prime factorization of 13720:
13720 = 2^3 × 5 × 7^2 × 11
Now, to make the quotient a perfect cube, we need to divide the given number by the smallest possible number that would result in an integer cube.
For a perfect cube, the exponents of all the prime factors must be divisible by 3.
The exponent of 2 is 3, which is already divisible by 3.
The exponent of 5 is 1, which is not divisible by 3.
The exponent of 7 is 2, which is not divisible by 3.
The exponent of 11 is 1, which is not divisible by 3.
Therefore, we need to divide 13720 by 2×5 = 10, which is the smallest number that contains all the prime factors and makes the exponent of each factor divisible by 3.
On dividing 13720 by 10, we get 1372, which is a perfect cube of 11.
Hence, the least number by which 13720 must be divided to get a perfect cube quotient is 10.
Therefore, the correct answer is option D) 5.
To find the least number by which 13720 must be divided to get a perfect cube quotient, we need to factorize the given number into its prime factors.
Prime factorization of 13720:
13720 = 2^3 × 5 × 7^2 × 11
Now, to make the quotient a perfect cube, we need to divide the given number by the smallest possible number that would result in an integer cube.
For a perfect cube, the exponents of all the prime factors must be divisible by 3.
The exponent of 2 is 3, which is already divisible by 3.
The exponent of 5 is 1, which is not divisible by 3.
The exponent of 7 is 2, which is not divisible by 3.
The exponent of 11 is 1, which is not divisible by 3.
Therefore, we need to divide 13720 by 2×5 = 10, which is the smallest number that contains all the prime factors and makes the exponent of each factor divisible by 3.
On dividing 13720 by 10, we get 1372, which is a perfect cube of 11.
Hence, the least number by which 13720 must be divided to get a perfect cube quotient is 10.
Therefore, the correct answer is option D) 5.
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What is the least number by which 13720 must be divided so that the quotient is a perfect cubea)4b)2c)7d)5Correct answer is option 'D'. Can you explain this answer?
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