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Question

Difficulty: EasyFactors, Multiples, and Prime Factorization

Consider the positive integer 9090. Which of the following statements regarding the factors and prime factorization of 9090 are correct?

  1. The prime factorization of 9090 is 21×32×512^1 \times 3^2 \times 5^1.Answer
  2. The total number of positive factors of 9090 is 1212.Answer
  3. C
    The number of positive even factors of 9090 is 88.
  4. D
    The sum of all distinct prime factors of 9090 is 1515.

Answer

The prime factorization of 90 is 2^1 × 3^2 × 5^1, and the total number of positive factors of 90 is 12.
The prime factorization of 90 is indeed 2^1 × 3^2 × 5^1. The total number of positive factors is calculated as (1+1)(2+1)(1+1) = 12, making both statement options correct.

Step-by-Step Solution

1
Find the prime factorization of 90.
90 = 2 × 45 = 2 × 3^2 × 5 = 2^1 × 3^2 × 5^1.
Decomposing 90 into prime base powers establishes the prime factorization.
2
Calculate the total number of positive factors using exponents.
Total factors = (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12.
Adding 1 to each prime exponent and multiplying gives the total factor count.
3
Verify even factors and prime factor sum.
Even factors = 1 × (2 + 1)(1 + 1) = 6. Distinct prime sum = 2 + 3 + 5 = 10.
Confirming additional properties disproves the incorrect statements.

Key Concept

Prime factorization and formulas for determining total and specific factor counts of an integer
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Topics

Factors, Multiples, and Prime Factorization

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