Question

The number 10,010 has how many positive integer factors? a. 31 b. 32 c. 33 d. 34 e. 35

Short answer

32

Step-by-step solution

Prime Factorization

Find the prime factorization of the number 10,010. Begin by dividing by the smallest prime numbers. 10,010 is even, so divide by 2:10,010 ÷ 2 = 5005.Next, 5005 is divisible by 5 (last digit is 5):5005 ÷ 5 = 1001.Observe 1001 = 7 × 143:1001 ÷ 7 = 143.Then, 143 can be further factorized by 11 (143 = 11 × 13):143 ÷ 11 = 13.Thus, we have the prime factorization: 10,010 = 2^1 × 5^1 × 7^1 × 11^1 × 13^1.

Determine Number of Factors

Use the formula for finding the number of positive integer factors of a number given its prime factorization. For a number with prime factorization \[N = p_1^{e_1} × p_2^{e_2} × \text{...} × p_k^{e_k} \]The number of factors is \[(e_1 + 1) × (e_2 + 1) × \text{...} × (e_k + 1).\]For 10,010, the prime factorization is 2^1 × 5^1 × 7^1 × 11^1 × 13^1, so we add 1 to each of the exponents and multiply:\[(1 + 1) × (1 + 1) × (1 + 1) × (1 + 1) × (1 + 1) = 2 × 2 × 2 × 2 × 2.\]

Calculate the Total

Calculate the product of the exponents plus one:\[2 × 2 × 2 × 2 × 2 = 32.\]Thus, the number 10,010 has 32 positive integer factors.

Key concepts

Prime Factorization

Prime factorization involves breaking down a number into its basic building blocks, which are prime numbers. Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. For example, the prime factors of the number 10,010 can be found as follows:

• Start by dividing 10,010 by the smallest prime number, which is 2. This gives us 5005.
• Next, 5005 can be divided by 5, resulting in 1001.
• Then, 1001 can be broken down further by 7 to give 143.
• Finally, divide 143 by 11, yielding 13 (which is also prime).

The factorization process stops here, and we list all the prime factors: 10,010 = 2 × 5 × 7 × 11 × 13. Each factor is raised to the power of 1.

Understanding prime factorization is crucial for various applications in mathematics, such as finding the greatest common divisor (GCD) or the least common multiple (LCM) of numbers.

Positive Integer Factors

Once you have the prime factorization of a number, you can determine the number of positive integer factors using a straightforward formula. If a number has the prime factorization: \[ N = p_1^{e_1} × p_2^{e_2} × \text{...} × p_k^{e_k} \] the number of factors is \[ (e_1 + 1) × (e_2 + 1) × \text{...} × (e_k + 1). \]

For 10,010, the prime factorization is 2^1 × 5^1 × 7^1 × 11^1 × 13^1. Applying the formula, we add 1 to each exponent and multiply the results together:
\[ (1 + 1) × (1 + 1) × (1 + 1) × (1 + 1) × (1 + 1) = 2 × 2 × 2 × 2 × 2 = 32. \] This means that 10,010 has 32 positive integer factors.

Understanding how to find and use positive integer factors is a key concept in number theory that will serve you well in many areas, from solving equations to optimizing algorithms.

GMAT Preparation

When preparing for the GMAT, mastering math concepts like prime factorization and positive integer factors is essential. These skills are part of the problem-solving and data sufficiency sections, where you need to think quickly and accurately.

Here are some tips for incorporating these concepts into your GMAT preparation:

• Practice regularly by solving various prime factorization problems.
• Understand the underlying principles to easily break down any number into its prime factors.
• Work on timed exercises to get comfortable with solving problems efficiently.
• Use resources like prep books, online courses, and practice tests to expose yourself to different types of factorization questions.
• Review detailed solutions to understand mistakes and learn better approaches.

Staying consistent and practicing smartly will greatly improve your confidence and ability to tackle GMAT math problems effectively.

Mathematical Formulae

Grasping mathematical formulae is a crucial aspect of solving any math problem, especially for exams like the GMAT. Knowing the specific formula to apply can save you valuable time and effort.

For finding the number of factors of a number, remember to use:
\[ (e_1 + 1) × (e_2 + 1) × \text{...} × (e_k + 1) \] when the prime factorization is: \[ N = p_1^{e_1} × p_2^{e_2} × \text{...} × p_k^{e_k}. \]

Other important formulas that can be useful for GMAT preparation include:

• \textbf{Area and Perimeter of Geometric Shapes:} Effective for geometry-related questions.
• \textbf{Quadratic Equations:} \[ ax^2 + bx + c = 0 \] helps in various algebraic problems.
• \textbf{Probability:} \[ P(A) = \frac{Number\text{ of favorable outcomes}}{Total\text{ number of outcomes}}. \] Important for data sufficiency problems.
• \textbf{Rates of Work and Mixtures:} Useful in word problems.

Memorizing and understanding when and how to apply these formulas will provide a strong foundation for tackling GMAT math problems efficiently and accurately.