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N is the smallest positive integer such that n/2 is a perfect square, n/3 is a perfect cube and n/5 is a perfect fifth power. If n is represented in its factorised form as p^a * q^b * r^c *., where p, q, r. . are all distinct prime numbers, then find the minimum sum a b c . (a) 31 (b) 37 (c) 54 (d) 56?
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N is the smallest positive integer such that n/2 is a perfect square, ...
To find the smallest positive integer N that satisfies the given conditions, we need to consider the prime factors of N. Let's break down the problem step by step:
Step 1: Find the common factors
Since N is divisible by 2, 3, and 5, we need to find the minimum powers of these prime numbers that make N a perfect square, a perfect cube, and a perfect fifth power, respectively.
Step 2: Prime factorize N
To find the value of N, we need to find the minimum powers of the prime factors that satisfy the given conditions. Let's assume the prime factorization of N is as follows: N = p^a * q^b * r^c * ...
Step 3: Determine the powers of prime factors
We know that N/2 is a perfect square, N/3 is a perfect cube, and N/5 is a perfect fifth power. Let's calculate the powers of the prime factors based on these conditions.
For N/2 to be a perfect square, the power of 2 in N should be even. Hence, a must be even.
For N/3 to be a perfect cube, the power of 3 in N should be a multiple of 3. Hence, b must be a multiple of 3.
For N/5 to be a perfect fifth power, the power of 5 in N should be a multiple of 5. Hence, c must be a multiple of 5.
Step 4: Find the minimum sum a + b + c
To minimize the sum a + b + c, we should choose the smallest values for a, b, and c that satisfy the conditions. The smallest even number for a is 2, the smallest multiple of 3 for b is 3, and the smallest multiple of 5 for c is 5.
Therefore, the minimum sum a + b + c = 2 + 3 + 5 = 10.
Step 5: Convert the sum to the given options
The given options are (a) 31, (b) 37, (c) 54, and (d) 56. None of these options match the minimum sum we calculated (10), so none of the given options is the correct answer.
Conclusion:
The minimum sum of the powers a + b + c is 10, which is not among the given options. Therefore, none of the given options is the correct answer.
Step 1: Find the common factors
Since N is divisible by 2, 3, and 5, we need to find the minimum powers of these prime numbers that make N a perfect square, a perfect cube, and a perfect fifth power, respectively.
Step 2: Prime factorize N
To find the value of N, we need to find the minimum powers of the prime factors that satisfy the given conditions. Let's assume the prime factorization of N is as follows: N = p^a * q^b * r^c * ...
Step 3: Determine the powers of prime factors
We know that N/2 is a perfect square, N/3 is a perfect cube, and N/5 is a perfect fifth power. Let's calculate the powers of the prime factors based on these conditions.
For N/2 to be a perfect square, the power of 2 in N should be even. Hence, a must be even.
For N/3 to be a perfect cube, the power of 3 in N should be a multiple of 3. Hence, b must be a multiple of 3.
For N/5 to be a perfect fifth power, the power of 5 in N should be a multiple of 5. Hence, c must be a multiple of 5.
Step 4: Find the minimum sum a + b + c
To minimize the sum a + b + c, we should choose the smallest values for a, b, and c that satisfy the conditions. The smallest even number for a is 2, the smallest multiple of 3 for b is 3, and the smallest multiple of 5 for c is 5.
Therefore, the minimum sum a + b + c = 2 + 3 + 5 = 10.
Step 5: Convert the sum to the given options
The given options are (a) 31, (b) 37, (c) 54, and (d) 56. None of these options match the minimum sum we calculated (10), so none of the given options is the correct answer.
Conclusion:
The minimum sum of the powers a + b + c is 10, which is not among the given options. Therefore, none of the given options is the correct answer.
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N is the smallest positive integer such that n/2 is a perfect square, n/3 is a perfect cube and n/5 is a perfect fifth power. If n is represented in its factorised form as p^a * q^b * r^c *., where p, q, r. . are all distinct prime numbers, then find the minimum sum a b c . (a) 31 (b) 37 (c) 54 (d) 56?
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