Understanding the Core Problem
The question asks for the minimum possible value for the sum \((p+q)\), where \(p\) and \(q\) are natural numbers (meaning \(p \ge 1\) and \(q \ge 1\)). The condition is that the expression \((p+q)^{p+q}\) must be perfectly divisible by 512.
Analyzing the Divisibility Condition
First, let's find the prime factorization of 512.
\(512 = 2 \times 256 = 2 \times 2^8 = 2^9\)
So, the condition is that \((p+q)^{p+q}\) must be divisible by \(2^9\).
Let \(N = p+q\). Since \(p\) and \(q\) are natural numbers, the smallest possible value for \(N\) is \(1+1 = 2\). The condition becomes \(N^N\) must be divisible by \(2^9\).
Relating Divisibility to Prime Factors
For \(N^N\) to be divisible by \(2^9\), the prime factorization of \(N\) must include the prime factor 2. This means \(N\) must be an even number.
Let the prime factorization of \(N\) be \(N = 2^k \cdot m\), where \(m\) is an odd integer and \(k \ge 1\) (since N is even). Then, the expression \(N^N\) can be written as:
\(N^N = (2^k \cdot m)^N = (2^k)^N \cdot m^N = 2^{k \cdot N} \cdot m^N\)
For \(N^N\) to be divisible by \(2^9\), the exponent of 2 in its prime factorization must be at least 9. Therefore, we need:
\(k \cdot N \ge 9\)
Finding the Least Value of N = (p+q)
We need to find the smallest integer \(N \ge 2\) such that \(N\) is even, and if \(N = 2^k \cdot m\) (with \(m\) odd), then \(k \cdot N \ge 9\). Let's test values:
- If \(N = 2\): \(N = 2^1 \cdot 1\). So, \(k=1\). \(k \cdot N = 1 \cdot 2 = 2\). Since \(2 < 9\), this doesn't work. (\(2^2 = 4\), not divisible by 512).
- If \(N = 4\): \(N = 2^2 \cdot 1\). So, \(k=2\). \(k \cdot N = 2 \cdot 4 = 8\). Since \(8 < 9\), this doesn't work. (\(4^4 = 256\), not divisible by 512).
- If \(N = 6\): \(N = 2^1 \cdot 3\). So, \(k=1\). \(k \cdot N = 1 \cdot 6 = 6\). Since \(6 < 9\), this doesn't work. (\(6^6 = 46656\), \(46656 / 512 \approx 91.1\), not divisible).
- If \(N = 8\): \(N = 2^3 \cdot 1\). So, \(k=3\). \(k \cdot N = 3 \cdot 8 = 24\). Since \(24 \ge 9\), this condition is satisfied. (\(8^8 = (2^3)^8 = 2^{24}\), which is clearly divisible by \(2^9\)).
Since we are looking for the least value of \((p+q)\), and \(N=8\) is the first value that satisfies the condition \(k \cdot N \ge 9\), it is the minimum possible value for \((p+q)\). We can easily find natural numbers \(p\) and \(q\) that sum to 8 (e.g., \(p=4, q=4\) or \(p=1, q=7\)).
Conclusion
The least value of \((p+q)\) such that \((p+q)^{p+q}\) is divisible by 512 is 8.
Similar Questions
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Let \(x = n(n+1)(n+2)\), where \(n\) is an even natural number. Which of the following statements is/are correct?
I. \(x\) is always divisible by 48.
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Select the answer using the code given below.
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Important Questions from Number System
What is the value of 1 2 + 2 2 + 3 2 + ......21 2 ?
Which sequence is correct to represent the hierarchical chain of number system?
(Where N - Natural Numbers
W - Whole Numbers
Q - Rational Numbers
Z - Integers)
The difference of the place value and the face value of 5 in 26549 is :
What must be added to 45680 to make it exactly divisible by 9?
How many zeroes are there at the end of the following product?
1 x 5 x 10 x 15 x 20 x 25 x 30 x 35 x 40 x 45 x 50 x 55 x 60
