What is the Smallest Prime Factor of 91?

When exploring number theory, one intriguing question arises: What is the smallest prime factor of 91? Understanding prime factors helps unlock deeper insights into divisibility, factorization, and properties of numbers—skills valuable in math, cryptography, and problem-solving.

Breaking Down 91: A Step-by-Step Approach

Understanding the Context

To find the smallest prime factor of 91, start by recalling that a prime factor is a prime number that divides 91 exactly without leaving a remainder. We analyze how 91 breaks down:

  1. Check divisibility by the smallest prime: 2
    91 is odd, so it is not divisible by 2.
    → Not a prime factor.

  2. Try the next prime: 3
    Add digits: 9 + 1 = 10. Since 10 is not divisible by 3, 91 is not divisible by 3.
    → Not a prime factor.

  3. Next prime: 5
    91 does not end in 0 or 5, so it’s not divisible by 5.
    → Not a prime factor.

Question:** What is the smallest prime factor of 91?

Key Insights

  1. Check divisibility by 7
    Perform division:
    91 ÷ 7 = 13
    Since 7 × 13 = 91, 7 divides 91 evenly.
    Both 7 and 13 are prime numbers.

The Answer: The Smallest Prime Factor of 91

Among the prime factors of 91—7 and 13—the smallest is 7.

Thus, the smallest prime factor of 91 is 7.

Why This Matters

Continue Reading

A stock decreases in value by 20% in the first year and then increases by 30% in the second year. If the initial value was $100, what is the value at the end of the second year?
First year decrease: \(100 \times 0.20 = 20\)
Value after first year: \(100 - 20 = 80\)

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Final Thoughts

Identifying prime factors is fundamental in simplifying fractions, solving equations, and understanding number behavior. For example, 91 factors into 7 × 13, meaning it’s a product of two primes—this classification is crucial in areas like RSA encryption, where composite numbers built from large primes form the basis of secure communication.

Summary

  • 91 is divisible only by 1, 7, 13, and 91.
  • Prime factors: 7 and 13.
  • The smallest prime factor is 7.

Mastering such foundational concepts strengthens mathematical reasoning and opens doors to advanced topics in science and technology.

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