mirror of
https://github.com/correl/euler.git
synced 2024-11-23 19:19:53 +00:00
72 lines
2.2 KiB
Python
72 lines
2.2 KiB
Python
|
"""Find the value of d < 1000 for which 1/d contains the longest recurring cycle.A unit fraction contains 1 in the numerator.
|
|||
|
|
|
|||
|
|
The decimal representation of the unit fractions with denominators 2 to 10 are given:
|
|||
|
|
1/2 = 0.5
|
|||
|
|
1/3 = 0.(3)
|
|||
|
|
1/4 = 0.25
|
|||
|
|
1/5 = 0.2
|
|||
|
|
1/6 = 0.1(6)
|
|||
|
|
1/7 = 0.(142857)
|
|||
|
|
1/8 = 0.125
|
|||
|
|
1/9 = 0.(1)
|
|||
|
|
1/10 = 0.1
|
|||
|
|
Where 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can be seen that 1/7 has a 6-digit recurring cycle.
|
|||
|
|
|
|||
|
|
Find the value of d < 1000 for which 1/d contains the longest recurring cycle in its decimal fraction part.
|
|||
|
|
"""
|
|||
|
|
|
|||
|
|
from decimal import Decimal
|
|||
|
|
from e003 import pfactor
|
|||
|
|
|
|||
|
|
def coprime(a, b):
|
|||
|
|
"""Are integers a and b coprime?"""
|
|||
|
|
|
|||
|
|
primes_a = pfactor(a)
|
|||
|
|
primes_b = pfactor(b)
|
|||
|
|
primes = [p for p in primes_a if p in primes_b]
|
|||
|
|
return bool(primes)
|
|||
|
|
|
|||
|
|
def period_length(numerator, denominator):
|
|||
|
|
"""Determine the period length of a non-terminating decimal"""
|
|||
|
|
|
|||
|
|
length = None
|
|||
|
|
numerator = Decimal(numerator)
|
|||
|
|
denominator = Decimal(denominator)
|
|||
|
|
if numerator == 1:
|
|||
|
|
if denominator == 1:
|
|||
|
|
return 1
|
|||
|
|
primes = pfactor(int(denominator))
|
|||
|
|
|
|||
|
|
"""
|
|||
|
|
From Wikipedia:
|
|||
|
|
Terminating decimals represent rational numbers of the form
|
|||
|
|
k/2^n5^m
|
|||
|
|
|
|||
|
|
However, a terminating decimal also has a representation as a
|
|||
|
|
repeating decimal, obtained by decreasing the final (nonzero)
|
|||
|
|
digit by one and appending an infinitely repeating sequence of
|
|||
|
|
nines. 1 = 0.999999... and 1.585 = 1.584999999... are two
|
|||
|
|
examples of this.
|
|||
|
|
"""
|
|||
|
|
if len([p for p in primes if p not in [2, 5]]) == 0:
|
|||
|
|
return 1
|
|||
|
|
|
|||
|
|
if coprime(denominator, 10):
|
|||
|
|
pass
|
|||
|
|
else:
|
|||
|
|
raise Exception('Non-reciprocal period detection not yet implemented')
|
|||
|
|
return length
|
|||
|
|
|
|||
|
|
def main():
|
|||
|
|
MAX = 10
|
|||
|
|
longest_cycle = 0
|
|||
|
|
for i in xrange((MAX), 1, -1):
|
|||
|
|
# The period of 1/k for integer k is always <20> k ? 1.
|
|||
|
|
if longest_cycle >= i:
|
|||
|
|
break
|
|||
|
|
|
|||
|
|
cycle = period_length(1, i)
|
|||
|
|
fraction = Decimal(1) / Decimal(i)
|
|||
|
|
print '1/{0} = {1} ({2})'.format(i, fraction, cycle), fraction.is_subnormal()
|
|||
|
|
if __name__ == '__main__':
|
|||
|
|
main()
|