Consider Euclid's Proof (By Refutation) that there are infinitely many primes :

Proof: Suppose that there were only finitely many prime numbers. Then, we could write them all down in a list as, {p1,p2,...pn}. Consider now their product plus one, giving the new number:

P = (p1 x p2 x.. x pn)+1

P is necessarily larger than any of the primes on our list. What remainder does this new number P leave when divided by any one of the pi for 1
≤
i
≤
n?

Question

Consider Euclid's Proof (By Refutation) that there are infinitely many primes :

Proof: Suppose that there were only finitely many prime numbers. Then, we could write them all down in a list as, {p1,p2,...pn}. Consider now their product plus one, giving the new number:

P = (p1 x p2 x.. x pn)+1

P is necessarily larger than any of the primes on our list.  What remainder does this new number P leave when divided by any one of the pi for 1
≤
i
≤
n?

BlackTom AI · Accepted Answer

For any listed prime pi, the product p1·p2·...·pn is divisible by pi, so adding 1 makes the remainder 1 when divided by pi.

1. Relevant concepts
- Division algorithm and remainders: when an integer A is divided by a positive integer d, the remainder r satisfies 0 ≤ r < d and A ≡ r (mod d).
- Modular arithmetic: A + B ≡ (A mod d) + (B mod d) (mod d).
- Euclid’s construction: P = p1·p2·...·pn + 1.

2. Step-by-step reasoning
Take any fixed pi from the list {p1, p2, ..., pn}. The product M = p1·p2·...·pn contains pi as a factor, so M is divisible by pi. Write P = M + 1. When dividing P by pi, consider P mod pi.

3. Intermediate calculations
- M ≡ 0 (mod pi) because pi | M.
- 1 ≡ 1 (mod pi).
- Therefore P ≡ M + 1 ≡ 0 + 1 ≡ 1 (mod pi).
Thus the remainder upon division of P by pi is 1. Example: with primes 2,3,5: M = 30, P = 31. 31 ÷ 2 leaves remainder 1; 31 ÷ 3 leaves remainder 1; 31 ÷ 5 leaves remainder 1.

4. Why the correct answer is correct
Because each pi divides the product M exactly, adding 1 guarantees that P leaves remainder 1 for every pi. This is the key observation in Euclid’s refutation: P cannot be divisible by any listed prime, so either P is a new prime or has prime factors outside the list.

5. Why the other options are incorrect
- "0": would mean pi divides P, but pi divides M, not M+1, so pi cannot divide P.
- "a number less than pi": while 1 is less than pi, this option is vague and not the exact remainder determined by the argument; the remainder is specifically 1.
- "a number greater than pi": impossible by the division algorithm because a remainder must be strictly less than the divisor pi.

6. Conclusion
The remainder is 1, confirming the provided correct answer: 1.

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