36 lines
1.1 KiB
Python
36 lines
1.1 KiB
Python
#!/usr/bin/env python3
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# Consider the fraction, n/d, where n and d are positive integers. If n<d and HCF(n,d)=1, it is called a reduced proper fraction.
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#
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# If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size, we get:
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#
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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, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8
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#
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# It can be seen that there are 21 elements in this set.
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#
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# How many elements would be contained in the set of reduced proper fractions for d ≤ 1,000,000?
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from projecteuler import sieve, phi, timing
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@timing
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def p072():
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N = 1000001
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count = 0
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primes = sieve(N)
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# For any denominator d, the number of reduced proper fractions is
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# the number of fractions n/d where gcd(n, d)=1, which is the definition
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# of Euler's Totient Function phi. It's sufficient to calculate phi for each
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# denominator and sum the value.
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for i in range(2, N):
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count = count + phi(i, primes)
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print('Project Euler, Problem 72')
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print(f'Answer: {int(count)}')
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if __name__ == '__main__':
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p072()
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