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In the proof of the infinitude of prime numbers, if we assume there are only \(n\) primes \(p_1, \dots, p_n\), the number \(M = p_1 p_2 \cdots p_n + 1\) is used because:
It is guaranteed to be a prime number itself.
It leaves a remainder of \(1\) when divided by any of the known primes \(p_k\).
It is always an even number.
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