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Ejercicio .) Enunciar el Teorema Fundamental de la Aritmética y utilizarlo para mostrar que es el número natural más chico que admite exactamente divisores positivos.) Hallar tal que el coeficiente que multiplica a en es .
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Q. Ejercicio .) Enunciar el Teorema Fundamental de la Aritmética y utilizarlo para mostrar que es el número natural más chico que admite exactamente divisores positivos.) Hallar tal que el coeficiente que multiplica a en es .
- State Fundamental Theorem of Arithmetic: State the Fundamental Theorem of Arithmetic and use it to show that is the smallest natural number with exactly positive divisors.The Fundamental Theorem of Arithmetic states that every integer greater than can be represented uniquely as a product of prime numbers, up to the order of the factors.The number of divisors is given by , which confirms that has exactly divisors.
- Expand Binomial Theorem: Expand using the binomial theorem to find the term containing . We need the term where the powers of add up to , which is the term with from raised to the th power and raised to the rd power. The binomial coefficient for this term is .
- Calculate Binomial Coefficient: Calculate the binomial coefficient .
- Write Term with : Write down the term containing . The term is . Substitute the binomial coefficient: . Simplify the powers of x: .
- Find Value of : Find the value of such that the coefficient of is . We need to equate the coefficient of in the term to , but we made a mistake in the previous step; we need the coefficient of , not .
