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

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All match. Thus, even if not strictly cubic, $ p(0) = 0 $. But perhaps we misread—maybe the problem says cubic but the data forces lower degree. Since the interpolation is unique, and the values sit on a quadratic, the answer is still $ p(0) = 0 $. Alternatively, suppose we assume $ p(x) $ is cubic: $ p(x) = ax^3 + 2x^2 + x $. Then $ p(1) = a + 2 + 1 = 3 \Rightarrow a = 0 $. So $ a = 0 $, same result.

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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