2018-09-28-104529.sagews

#Problem 1 - Count the number of divisors of a positive integer n.
def num_div(n):
    i=1
    divisors=0
    while i<=n:
        if n%i==0:
            divisors=divisors+1
        i=i+1
    return divisors

num_div(10)

4

num_div(12)

6

num_div(10000)

25

def primetest(n):
#true if n is prime
    if n==1:
        return false
    i=2
    while i<=n^(0.5):
        if n%i==0:
            return False
        i=i+1
    return True

#Problem 2 - Determines if there is a prime in the interval [n,n+13]
def prime_interval(n):
    i=n
    numprimes=0
    while n<=i<=(n+13):
        if primetest(i):
            numprimes=numprimes+1
        i=i+1
    if numprimes>0:
        return True
    if numprimes==o:
        return False

prime_interval(1025)

True

#Problem 3 - Takes two positive integers and returns the maximum of the two.
def int_max(a,b):
    if a>b:
        return a
    if b>a:
        return b
    if a==b:
        return "Error: a=b"

int_max(10,3)

10

int_max(10,10)

'Error: a=b'

int_max(100, 1253323)

1253323

#Problem 4 - Takes a list of numbers and returns the maximum value.
def list_max(L):
    if len(L)==0:
        print 'Error: no elements'
        return
    if len(L)==1:
        return L[0]
    currentmax=L[0]
    i=1
    while i<=len(L)-1:
        b=L[i]
        if b>currentmax:
            currentmax=b
        i=i+1
    return currentmax

L=[1,64,23,76,15,26,100]

list_max(L)

100

def trials(n):
    i=1
    success=0
    while i<=n:
        if experiment(n):
            success=success+1
        i=i+1
    return (success+0.0)/n

#Problem 5 - True if a and b are relatively prime.
def experiment(n):
    a=randint(1,n)
    b=randint(1,n)
    if gcd(a,b)==1:
        return True

trials(100)

0.590000000000000

trials(1000)

0.592000000000000

trials(10^4)

0.609800000000000

trials(10^5)

0.609190000000000

trials(10^6)

0.607294000000000

#Problem 6
def experiment(n):
    i=n
    while n<=i<=(n+14):
        if is_prime(i):
            return True
        i=i+1
    return False

#Problem 7 - True if the number of heads in n random coin tosses is at least 7.
def coinflip(n):
    i=1
    numheads=0
    while i<=n:
        toss=randint(0,1) #0 = heads, 1 = tails
        if toss==0:
            numheads=numheads+1
        if toss==1:
            numheads=numheads
        i=i+1
    if numheads>=5:
        return True
    if numheads<5:
        return False

coinflip(10)

False

def fliptrials(n):
    i=1
    success=0
    while i<=n:
        if coinflip(10):
            success=success+1
        i=i+1
    return (success+0.0)/n

fliptrials(100000)

0.623570000000000

N(319/512)

0.623046875000000