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46 lines
1.2 KiB
Python
46 lines
1.2 KiB
Python
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"""
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Project Euler Problem 72: https://projecteuler.net/problem=72
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Consider the fraction, n/d, where n and d are positive integers. If n<d and HCF(n,d)=1,
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it is called a reduced proper fraction.
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If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size,
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we get:
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1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2,
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4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8
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It can be seen that there are 21 elements in this set.
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How many elements would be contained in the set of reduced proper fractions
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for d ≤ 1,000,000?
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"""
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def solution(limit: int = 1000000) -> int:
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"""
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Return the number of reduced proper fractions with denominator less than limit.
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>>> solution(8)
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21
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>>> solution(1000)
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304191
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"""
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primes = set(range(3, limit, 2))
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primes.add(2)
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for p in range(3, limit, 2):
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if p not in primes:
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continue
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primes.difference_update(set(range(p * p, limit, p)))
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phi = [float(n) for n in range(limit + 1)]
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for p in primes:
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for n in range(p, limit + 1, p):
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phi[n] *= 1 - 1 / p
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return int(sum(phi[2:]))
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if __name__ == "__main__":
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print(f"{solution() = }")
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