Question

Here is a table of remainders when powers of 10 are divided by 2020:

k	10k	Remainder
0	1	1
1	10	10
2	100	100
3	1000	1000
4	10000	1920
5	100000	1020
6	1000000	100
7	10000000	1000
8	100000000	1920
9	1000000000	1020

We see that the remainders repeat every four steps (period 4), with two exceptions at the beginning, 1 and 10. We will call a sequence that repeats with period 4, with two exceptions at the beginning, a fortuitous sequence (four-two-itous). Sequences that have periods smaller than four (e.g. sequences that repeat every two steps) do not count as fortuitous.

1. In addition to 2020, for what other values of m is the sequence of remainders when 10^k is divided by m a fortuitous sequence?
2. In addition to 10, for what other values of a is the sequence of remainders when a^k is divided by 2020 a fortuitous sequence?

Answer 1

def a():
m = int(input("In addition to 2020, for what other values of m is the sequence of remainders when 10k is divided by m a fortuitous sequence? So enter upper limit m = "))
for n in range(2,m):
r = []
for i in range(10):
r.append(pow(10,i)%n)
"""if (r!=0):
print(r)"""
if (r[2:6] == r[6:10]):
print("Remainder of powers of 10 dividing by ",n," is a fortuitous sequence")
def b():
a = int(input("In addition to 10, for what other values of a is the sequence of remainders when ak is divided by 2020 a fortuitous sequence? So enter upper limit a = "))
n = 2020
for b in range(2,a):
r = []
for i in range(a):
r.append(pow(b,i)%n)
"""if (r!=0):
print(r)"""
if (r[2:6] == r[6:10]):
print("Remainder of powers of",b,"dividing by ",n," is a fortuitous sequence")
ch = input("You want answer for a or b please enter a or b only ")
if(ch=="a"):
a()
elif(ch=="b"):
b()

"""Above is python code and Output is here

(a) m between(2,10000)

You want answer for a or b please enter a or b only : a
In addition to 2020, for what other values of m is the sequence of remainders when 10k is divided by m a fortuitous sequence? So enter upper limit m = 10000
Remainder of powers of 10 dividing by  2  is a fortuitous sequence
Remainder of powers of 10 dividing by  3  is a fortuitous sequence
Remainder of powers of 10 dividing by  4  is a fortuitous sequence
Remainder of powers of 10 dividing by  5  is a fortuitous sequence
Remainder of powers of 10 dividing by  6  is a fortuitous sequence
Remainder of powers of 10 dividing by  9  is a fortuitous sequence
Remainder of powers of 10 dividing by  10  is a fortuitous sequence
Remainder of powers of 10 dividing by  11  is a fortuitous sequence
Remainder of powers of 10 dividing by  12  is a fortuitous sequence
Remainder of powers of 10 dividing by  15  is a fortuitous sequence
Remainder of powers of 10 dividing by  18  is a fortuitous sequence
Remainder of powers of 10 dividing by  20  is a fortuitous sequence
Remainder of powers of 10 dividing by  22  is a fortuitous sequence
Remainder of powers of 10 dividing by  25  is a fortuitous sequence
Remainder of powers of 10 dividing by  30  is a fortuitous sequence
Remainder of powers of 10 dividing by  33  is a fortuitous sequence
Remainder of powers of 10 dividing by  36  is a fortuitous sequence
Remainder of powers of 10 dividing by  44  is a fortuitous sequence
Remainder of powers of 10 dividing by  45  is a fortuitous sequence
Remainder of powers of 10 dividing by  50  is a fortuitous sequence
Remainder of powers of 10 dividing by  55  is a fortuitous sequence
Remainder of powers of 10 dividing by  60  is a fortuitous sequence
Remainder of powers of 10 dividing by  66  is a fortuitous sequence
Remainder of powers of 10 dividing by  75  is a fortuitous sequence
Remainder of powers of 10 dividing by  90  is a fortuitous sequence
Remainder of powers of 10 dividing by  99  is a fortuitous sequence
Remainder of powers of 10 dividing by  100  is a fortuitous sequence
Remainder of powers of 10 dividing by  101  is a fortuitous sequence
Remainder of powers of 10 dividing by  110  is a fortuitous sequence
Remainder of powers of 10 dividing by  132  is a fortuitous sequence
Remainder of powers of 10 dividing by  150  is a fortuitous sequence
Remainder of powers of 10 dividing by  165  is a fortuitous sequence
Remainder of powers of 10 dividing by  180  is a fortuitous sequence
Remainder of powers of 10 dividing by  198  is a fortuitous sequence
Remainder of powers of 10 dividing by  202  is a fortuitous sequence
Remainder of powers of 10 dividing by  220  is a fortuitous sequence
Remainder of powers of 10 dividing by  225  is a fortuitous sequence
Remainder of powers of 10 dividing by  275  is a fortuitous sequence
Remainder of powers of 10 dividing by  300  is a fortuitous sequence
Remainder of powers of 10 dividing by  303  is a fortuitous sequence
Remainder of powers of 10 dividing by  330  is a fortuitous sequence
Remainder of powers of 10 dividing by  396  is a fortuitous sequence
Remainder of powers of 10 dividing by  404  is a fortuitous sequence
Remainder of powers of 10 dividing by  450  is a fortuitous sequence
Remainder of powers of 10 dividing by  495  is a fortuitous sequence
Remainder of powers of 10 dividing by  505  is a fortuitous sequence
Remainder of powers of 10 dividing by  550  is a fortuitous sequence
Remainder of powers of 10 dividing by  606  is a fortuitous sequence
Remainder of powers of 10 dividing by  660  is a fortuitous sequence
Remainder of powers of 10 dividing by  825  is a fortuitous sequence
Remainder of powers of 10 dividing by  900  is a fortuitous sequence
Remainder of powers of 10 dividing by  909  is a fortuitous sequence
Remainder of powers of 10 dividing by  990  is a fortuitous sequence
Remainder of powers of 10 dividing by  1010  is a fortuitous sequence
Remainder of powers of 10 dividing by  1100  is a fortuitous sequence
Remainder of powers of 10 dividing by  1111  is a fortuitous sequence
Remainder of powers of 10 dividing by  1212  is a fortuitous sequence
Remainder of powers of 10 dividing by  1515  is a fortuitous sequence
Remainder of powers of 10 dividing by  1650  is a fortuitous sequence
Remainder of powers of 10 dividing by  1818  is a fortuitous sequence
Remainder of powers of 10 dividing by  1980  is a fortuitous sequence
Remainder of powers of 10 dividing by  2020  is a fortuitous sequence
Remainder of powers of 10 dividing by  2222  is a fortuitous sequence
Remainder of powers of 10 dividing by  2475  is a fortuitous sequence
Remainder of powers of 10 dividing by  2525  is a fortuitous sequence
Remainder of powers of 10 dividing by  3030  is a fortuitous sequence
Remainder of powers of 10 dividing by  3300  is a fortuitous sequence
Remainder of powers of 10 dividing by  3333  is a fortuitous sequence
Remainder of powers of 10 dividing by  3636  is a fortuitous sequence
Remainder of powers of 10 dividing by  4444  is a fortuitous sequence
Remainder of powers of 10 dividing by  4545  is a fortuitous sequence
Remainder of powers of 10 dividing by  4950  is a fortuitous sequence
Remainder of powers of 10 dividing by  5050  is a fortuitous sequence
Remainder of powers of 10 dividing by  5555  is a fortuitous sequence
Remainder of powers of 10 dividing by  6060  is a fortuitous sequence
Remainder of powers of 10 dividing by  6666  is a fortuitous sequence
Remainder of powers of 10 dividing by  7575  is a fortuitous sequence
Remainder of powers of 10 dividing by  9090  is a fortuitous sequence
Remainder of powers of 10 dividing by  9900  is a fortuitous sequence
Remainder of powers of 10 dividing by  9999  is a fortuitous sequence

(b) a between (2,1000)

You want answer for a or b please enter a or b only : b
In addition to 10, for what other values of a is the sequence of remainders when ak is divided by 2020 a fortuitous sequence? So enter upper limit a = 1000
Remainder of powers of 10 dividing by  2020  is a fortuitous sequence
Remainder of powers of 91 dividing by  2020  is a fortuitous sequence
Remainder of powers of 100 dividing by  2020  is a fortuitous sequence
Remainder of powers of 101 dividing by  2020  is a fortuitous sequence
Remainder of powers of 102 dividing by  2020  is a fortuitous sequence
Remainder of powers of 111 dividing by  2020  is a fortuitous sequence
Remainder of powers of 192 dividing by  2020  is a fortuitous sequence
Remainder of powers of 201 dividing by  2020  is a fortuitous sequence
Remainder of powers of 202 dividing by  2020  is a fortuitous sequence
Remainder of powers of 203 dividing by  2020  is a fortuitous sequence
Remainder of powers of 212 dividing by  2020  is a fortuitous sequence
Remainder of powers of 293 dividing by  2020  is a fortuitous sequence
Remainder of powers of 302 dividing by  2020  is a fortuitous sequence
Remainder of powers of 303 dividing by  2020  is a fortuitous sequence
Remainder of powers of 304 dividing by  2020  is a fortuitous sequence
Remainder of powers of 313 dividing by  2020  is a fortuitous sequence
Remainder of powers of 394 dividing by  2020  is a fortuitous sequence
Remainder of powers of 403 dividing by  2020  is a fortuitous sequence
Remainder of powers of 404 dividing by  2020  is a fortuitous sequence
Remainder of powers of 405 dividing by  2020  is a fortuitous sequence
Remainder of powers of 414 dividing by  2020  is a fortuitous sequence
Remainder of powers of 495 dividing by  2020  is a fortuitous sequence
Remainder of powers of 504 dividing by  2020  is a fortuitous sequence
Remainder of powers of 505 dividing by  2020  is a fortuitous sequence
Remainder of powers of 506 dividing by  2020  is a fortuitous sequence
Remainder of powers of 515 dividing by  2020  is a fortuitous sequence
Remainder of powers of 596 dividing by  2020  is a fortuitous sequence
Remainder of powers of 605 dividing by  2020  is a fortuitous sequence
Remainder of powers of 606 dividing by  2020  is a fortuitous sequence
Remainder of powers of 607 dividing by  2020  is a fortuitous sequence
Remainder of powers of 616 dividing by  2020  is a fortuitous sequence
Remainder of powers of 697 dividing by  2020  is a fortuitous sequence
Remainder of powers of 706 dividing by  2020  is a fortuitous sequence
Remainder of powers of 707 dividing by  2020  is a fortuitous sequence
Remainder of powers of 708 dividing by  2020  is a fortuitous sequence
Remainder of powers of 717 dividing by  2020  is a fortuitous sequence
Remainder of powers of 798 dividing by  2020  is a fortuitous sequence
Remainder of powers of 807 dividing by  2020  is a fortuitous sequence
Remainder of powers of 808 dividing by  2020  is a fortuitous sequence
Remainder of powers of 809 dividing by  2020  is a fortuitous sequence
Remainder of powers of 818 dividing by  2020  is a fortuitous sequence
Remainder of powers of 899 dividing by  2020  is a fortuitous sequence
Remainder of powers of 908 dividing by  2020  is a fortuitous sequence
Remainder of powers of 909 dividing by  2020  is a fortuitous sequence
Remainder of powers of 910 dividing by  2020  is a fortuitous sequence
Remainder of powers of 919 dividing by  2020  is a fortuitous sequence