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Ejercicio 1. Enunciar el Teorema Fundamental de la Aritmética y utilizarlo para mostrar que 48 es el número natural más chico que admite exactamente 10 divisores positivos.

Ejercicio 11.\newline11) Enunciar el Teorema Fundamental de la Aritmética y utilizarlo para mostrar que 4848 es el número natural más chico que admite exactamente 1010 divisores positivos.

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Q. Ejercicio 11.\newline11) Enunciar el Teorema Fundamental de la Aritmética y utilizarlo para mostrar que 4848 es el número natural más chico que admite exactamente 1010 divisores positivos.
  1. State Theorem: State the Fundamental Theorem of Arithmetic: Every integer greater than 11 can be uniquely factored into prime numbers.
  2. Factor 4848: Factor 4848 into prime factors: 48=24×3148 = 2^4 \times 3^1.
  3. Calculate Divisors: Calculate the number of divisors using the formula (a+1)(b+1)…(a+1)(b+1)\ldots where a,b,…a, b, \ldots are the exponents of the prime factors: (4+1)(1+1)=5×2=10(4+1)(1+1) = 5 \times 2 = 10 divisors.
  4. Check Smallest Number: Check if 4848 is the smallest number with 1010 divisors: Test smaller numbers with the same number of divisors.
  5. Realize Smallest Number: Realize that 32=2532 = 2^5 has only 66 divisors, and 36=22×3236 = 2^2 \times 3^2 has 99 divisors. Since 4848 is the next number with more divisors, it is the smallest with 1010 divisors.

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