def p3(N=600851475143):
"""P3: What is the largest prime factor of the number 600851475143?
"""
# First attempt was to use a prime sieve first and then filter
# the factors, but my naive prime sieve (Eratosthenes)
# implementation ran into performance issues. This version uses
# the Pollard-Rho-Brent algorithm -- very, very efficient.
return max(nttools.pfactor(N))
def p5(N=20):
"""What is the smallest positive number that is evenly divisible
by all of the numbers from 1 to 20?
"""
# Factors of smallest number evenly divisible by every number
# from 1 to N given as follows: from all numbers between 2 and
# N, take the maximum multiplicity (n_i) of each factor (i).
fcounters = [Counter(nttools.pfactor(N=n)) for n in range(2, N+1)]
#could also have used pfactorGen(n)
factors = Counter()
for fc in fcounters:
for k, v in fc.items():
factors[k] = v if v > factors[k] else factors[k]
return reduce(op.mul, [k**v for k, v in factors.items()], 1)
