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Distributive Property of Rational Numbers

Suppose 3k× b...

Question

Suppose 3^k × b² = 6⁴ for some positive integers k, b. Find all possible values of k + b.

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Solution

Thus, we get .

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det ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} n \sum k = 0 k & n \sum k = 0^{n} C_{k} k^{2} n \sum k = 0^{n} C_{k} k & n \sum k = 0^{n} C_{k} 3^{k} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = 0

hold for some positive integer

n . Then n \sum k = 0 \frac{^{n} C_{k}}{k + 1}

b

is a positive integer such that

\frac{2^{4}}{b^{2}}

is also an integer. What are the possible values of

b

Suppose b is a positive integer such that is also an integer. What are the possible values of b?

det ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} n \sum k = 0 k & n \sum k = 0^{n} C_{k} k^{2} n \sum k = 0^{n} C_{k} k & n \sum k = 0^{n} C_{k} 3^{k} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = 0

hold for some positive integer

n . Then n \sum k = 0 \frac{^{n} C_{k}}{k + 1}

{log}_{a} b + {log}_{b} a = c

. The smallest possible integer value of

c

a, b > 1

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