Questions: Divisibility, Primes, and Fundamental Theorem of Arithmetic

Question 1

A student claims: '1 is prime because it has exactly two divisors: 1 and itself — and 1 = itself, so the two divisors are the same.' What is wrong with this reasoning?

• A. Nothing is wrong — 1 is actually prime by this definition
• B. The definition of prime requires exactly two *distinct* positive divisors. 1 has only one distinct positive divisor (which is 1 itself), so it has exactly one divisor, not two
• C. 1 is prime but only in the integers, not in other number systems
• D. The student is right about the divisors, but 1 is excluded from primes because it is not an integer greater than 1

Answer: (B)

Primes are defined as integers greater than 1 with exactly two distinct positive divisors: 1 and themselves. For any prime p > 1, these two divisors are different: 1 ≠ p. For the number 1, the only positive divisor is 1 itself — there is only one distinct divisor. So 1 has exactly one positive divisor, not two, and fails the prime definition. The real reason this matters: if 1 were prime, prime factorization would lose uniqueness (12 = 2²×3 = 1×2²×3 = 1²×2²×3, etc.), destroying the Fundamental Theorem of Arithmetic.

Question 2

Which part of the Fundamental Theorem of Arithmetic is mathematically non-trivial and requires careful proof?

• A. Existence — proving that every integer greater than 1 has at least one prime factorization
• B. Uniqueness — proving that the prime factorization is the same regardless of how you factor the number
• C. Both parts are equally trivial and follow immediately from the definition of prime
• D. Neither part requires proof — the FTA is an axiom of arithmetic

Answer: (B)

Existence is straightforward: if n > 1 is not prime, it factors as n = ab with 1 < a, b < n. Apply the same reasoning to a and b, and by strong induction (the factors decrease each time), you eventually reach all primes. Uniqueness is the deep part: you must show that no integer can be written as two genuinely different products of primes. The proof requires Euclid's lemma (if p is prime and p | ab, then p | a or p | b), which itself depends on properties of GCD. Uniqueness is also what fails in other algebraic systems like Z[√-5], where 6 = 2×3 = (1+√-5)(1-√-5) — two distinct factorizations into 'irreducibles.'

Question 3

The number 1 is not considered prime because the convention excluding it is arbitrary — mathematicians simply chose not to include it.

• T. True
• F. False

Answer: False

The exclusion of 1 is not arbitrary — it is necessary to preserve the uniqueness part of the Fundamental Theorem of Arithmetic. If 1 were prime, every integer would have infinitely many prime factorizations (just multiply by 1 repeatedly: 12 = 2²×3 = 1×2²×3 = 1²×2²×3 = ...). The FTA's uniqueness statement would collapse. Definitions in mathematics are chosen to make theorems work cleanly; excluding 1 from primes is a principled decision that preserves one of number theory's foundational results.

Question 4

The GCD of two integers can be computed from their prime factorizations by taking the minimum power of each common prime factor.

• T. True
• F. False

Answer: True

Yes — this is a direct application of the FTA. For example: 360 = 2³×3²×5 and 504 = 2³×3²×7. The common prime factors are 2 and 3. Taking the minimum exponent for each: min(3,3)=3 for 2 and min(2,2)=2 for 3. So gcd(360, 504) = 2³×3² = 8×9 = 72. The prime 5 appears only in 360, and 7 only in 504, so neither contributes to the GCD. This factorization-based method is conceptually clear, but for large numbers the Euclidean algorithm is far faster since factoring large integers is computationally hard.

Question 5

Why does the definition of prime numbers exclude 1, and what would break in arithmetic if 1 were classified as prime?

Model answer: 1 is excluded to preserve the uniqueness of prime factorization (the Fundamental Theorem of Arithmetic). If 1 were prime, every integer would have infinitely many factorizations: 12 = 2²×3 = 1×2²×3 = 1⁷×2²×3 = ... The theorem states that factorization is unique 'up to order of factors' — with 1 as a prime, there would be infinitely many ways to factor any integer, making the theorem false. The exclusion is a principled definition choice to make the FTA hold.

This illustrates a general principle in mathematics: definitions are not arbitrary — they are engineered to make theorems clean and powerful. The 'right' definition of prime is the one that makes the FTA and related results work. In more abstract algebra (ring theory), a 'prime' and an 'irreducible' are formally distinguished, and the FTA holds in rings called 'unique factorization domains' (UFDs). The integers Z are the canonical example of a UFD, and excluding 1 from the primes is part of what makes them one.