This problem has a cool name

Computer Science Level 3

What is the largest integer $n<1000$ such that the decimal representation of $2^n$ does not contain 0 as a digit?

Adapted from a book by J. M. de Koninck.

The answer is 86.

8 solutions

Brock Brown
Jul 1, 2015

Python 3.3:

n = 999
while '0' in str(2**n):
    n -= 1
print("Answer:", n)

Nice solution! +1

Arulx Z - 5 years, 11 months ago

Chew-Seong Cheong
Jun 30, 2015

My Python program is as follows:

for i in range(1000):
    n = 999 - i
    s = str(2**n)
    z = 0
    for j in range(len(s)):
        if s[j] == '0':
            z += 1
        if z == 0:
            print n
            break

Moderator note:

Good, standard solution.

is there alternative way to do it without Python program ?

Syed Baqir - 5 years, 11 months ago

Arulx Z
Jun 29, 2015

lar = None
for x in xrange(1, 999):
    if '0' not in str(2**x):
        lar = x
print lar

Moderator note:

Any particular reason to initialize lar to None than zero?

Any idea if there is a larger power of 2 which doesn't contain a 0?

Calvin Lin Staff - 5 years, 11 months ago

I am sorry but I have no idea on how to find a larger power of two which doesn't contain a 0. I have opened a note here to start a discussion.

Also, there is no specific reason behind why I initialized lar to None .

Arulx Z - 5 years, 11 months ago

Apparently it's an open problem in number theory

Arulx Z - 5 years, 7 months ago

Rushikesh Jogdand
Jun 24, 2016

index=0
for i in range(0,1001):
    if not '0' in list(set(list(str(2**i)))):
        index=i
print(index)

Kamala Ramakrishnan
Apr 8, 2016

VB.Net Solution:

Make sure to import System.Numerics.

Sub Main()

    For n = 1000 To 1 Step -1

        If Not ContainsZero(BigInteger.Pow(2, n)) Then

            Console.WriteLine(n)

        End If

    Next

    Console.ReadLine()

End Sub

Function ContainsZero(Number As BigInteger) As Boolean

    Dim a As String = Number.ToString

    For n = 0 To a.Length - 1

        If a(n) = "0" Then

            Return True

        End If

    Next

    Return False

End Function

Thank you for your solution! =D

Pi Han Goh - 5 years, 2 months ago

Rohan Gupta
Jul 21, 2015

My python 2.7 program:
for i in range(1000):
k=2**i
if "0" in str(k):
pass
else:
print i

Chandrachur Banerjee
Jul 21, 2015

My colourful layman's solution : y

Bill Bell
Jul 5, 2015

Check for occurrences of zero digits in all powers of two less than one thousand, and avoid exponentiating individual powers of two, which is expensive.

This problem has a cool name

Adapted from a book by J. M. de Koninck.

The answer is 86.

8 solutions

Moderator note:

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