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A prime has exactly two positive divisors: one and itself. Trial division only needs to reach √v, because any factor larger than that pairs with a smaller one already tested.
Primality by trial division
test divisibility by 2, 3, then 6k ± 1 up to √v
Yes — it is the largest four-digit prime.
It has only one divisor, and treating it as prime would break the uniqueness of prime factorisation.