Check if a number N starts with 1 in b-base

Given a number N and base b if N in base b representation starts with 1 print Yes else print No 

Examples :

Input : n = 6, b = 4
Output : Yes
6 can be written as  in base 4so answer is Yes as it starts with 1Input : n = 24, b = 2Output : Yes24 can be written as  in base 2so answer is Yes as it starts with 1Input :  n = 24, b = 7Output : No24 can be written as  in base 7so answer is No as it starts with 3        

When a number N is represented in base ‘b’ it gets converted to m+1 length sequence [Tex]b_{m-1}      [/Tex]…..which implies *+ *…..+*= N 

The smallest number in base b and starting with ‘1’ i.e. 100..00 and m+1 digits in base is 
and largest number is 2*-1.So N should lie in this range. 
<= N <= 2*-1 
Now another thing to notice is that m cannot exceed floor((N)) because when we represent any number in base-2 it gets converted into a sequence of only 1s and 0s so the length of this sequence will always be greater than any other base representation and its length will be equal to floor((N))+1. 

So to check for a given base ‘b’ if N starts with 1 or not we will traverse from m = 1 to m = floor((N)) and check if for any m N lies in the range <= N <= 2*-1 or not and accordingly print “Yes” or “No”. 

C++

// C++ program to check if number starts with
// one in base b representation
#include <bits/stdc++.h>
using namespace std;

// Returns true if n starts with  1 in
// base b representation
bool CheckIfstartsWithOne(int n, int b)
{
    // highest m can be log2(n)
    int m = log2(n);
    for (int i = 1; i <= m; i++) {
        // if b^m <= N <= 2*b^m - 1,
        // return true
        if (n >= pow(b, i) && n <= 2 * pow(b, i) - 1)
            return true;
    }
    return false;
}

// printing yes or no
void printYesORno(int n, int b)
{
    if (CheckIfstartsWithOne(n, b) == true)
        cout << "Yes" << endl;
    else if (CheckIfstartsWithOne(n, b) == false)
        cout << "No" << endl;
}

// driver function
int main()
{
    printYesORno(6, 4);
    printYesORno(24, 2);
    printYesORno(24, 7);
    printYesORno(24, 15);
}

Java

// Java program to check if number starts with
// one in base b representation

class GFG {

    // Returns true if n starts with 1 in
    // base b representation
    static boolean CheckIfstartsWithOne(int n, int b)
    {

        // highest m can be log2(n)
        int m = (int)(Math.log10(n) / Math.log10(2));
        for (int i = 1; i <= m; i++) {

            // if b^m <= N <= 2*b^m - 1,
            // return true
            if (n >= (int)Math.pow(b, i) &&
                n <= 2 * (int)Math.pow(b, i) - 1)
                return true;
        }

        return false;
    }

    // Driver method
    public static void main(String args[])
    {
        System.out.println(
           CheckIfstartsWithOne(6, 4) ? "Yes" : "No");

        System.out.println(
           CheckIfstartsWithOne(24, 2) ? "Yes" : "No");

        System.out.println(
           CheckIfstartsWithOne(24, 7) ? "Yes" : "No");

        System.out.println(
           CheckIfstartsWithOne(24, 15) ? "Yes" : "No");
    }
}

Python3

# Python3 program to check
# if number starts with one
# in base b representation
import math

# Returns true if n
# starts with 1 in
# base b representation
def CheckIfstartsWithOne(n, b):

    # highest m can be log2(n)
    m = (int)(math.log2(n));
    for i in range (1, m + 1):

        # if b^m <= N <= 2*b^m - 1,
        #return true
        x = (int)(math.pow(b, i));
        if n >= x and n <= 2 * x - 1:
            return 1;
    return 0;

# printing yes or no
def printYesORno(n, b):
    if CheckIfstartsWithOne(n, b) == 1:
        print("Yes");
    if CheckIfstartsWithOne(n, b) == 0:
        print("No");

# Driver Code
printYesORno(6, 4);
printYesORno(24, 2);
printYesORno(24, 7);
printYesORno(24, 15);

# This code is contributed by mits.

C#

// C# program to check if number starts with
// one in base b representation
using System;

class GFG{

// Returns true if n starts with 1 in
// base b representation
static bool CheckIfstartsWithOne(int n, int b)
{

    // highest m can be log2(n)
    int m = (int)(Math.Log10(n) / Math.Log10(2));
    for(int i = 1; i <= m; i++)
    {

        // if b^m <= N <= 2*b^m - 1,
        // return true
        if (n >= (int)Math.Pow(b, i) &&
            n <= 2 * (int)Math.Pow(b, i) - 1)
            return true;
    }

    return false;
}

// Driver code
public static void Main(String []args)
{
    Console.WriteLine(
       CheckIfstartsWithOne(6, 4) ? "Yes" : "No");

    Console.WriteLine(
       CheckIfstartsWithOne(24, 2) ? "Yes" : "No");

    Console.WriteLine(
       CheckIfstartsWithOne(24, 7) ? "Yes" : "No");

    Console.WriteLine(
       CheckIfstartsWithOne(24, 15) ? "Yes" : "No");
}
}

// This code is contributed by Princi Singh

PHP

<?php
// PHP program to check if
// number starts with one
// in base b representation

// Returns true if n starts
// with 1 in base b representation
function CheckIfstartsWithOne($n, $b)
{
    // highest m can be log2(n)
    $m = log($n, 2);
    for ($i = 1; $i <= $m; $i++)
    {
        // if b^m <= N <= 2*b^m - 1,
        // return true
        if ($n >= pow($b, $i) &&
                $n <= 2 * pow($b, $i) - 1)
            return true;
    }
    return false;
}

// printing yes or no
function printYesORno($n, $b)
{
    if (CheckIfstartsWithOne($n, $b) == true)
        echo "Yes" ,"\n";
    else if (CheckIfstartsWithOne($n, $b) == false)
        echo "No" ,"\n";
}

// Driver Code
printYesORno(6, 4);
printYesORno(24, 2);
printYesORno(24, 7);
printYesORno(24, 15);

// This code is contributed by ajit
?>

Javascript

<script>
    // Javascript program to check if number starts
    // with one in base b representation

    // Returns true if n starts with
    // 1 in base b representation
    function CheckIfstartsWithOne(n, b)
    {

        // highest m can be log2(n)
        let m = parseInt(Math.log10(n) / Math.log10(2), 10);

        for (let i = 1; i <= m; i++)
        {

            // if b^m <= N <= 2*b^m - 1,
            // return true
            if (n >= Math.pow(b, i) && n <= 2 * Math.pow(b, i) - 1)
                return true;
        }

        return false;
    }

    document.write(CheckIfstartsWithOne(6, 4) ? "Yes" + "</br>" : "No" + "</br>");

    document.write(CheckIfstartsWithOne(24, 2) ? "Yes" + "</br>" : "No" + "</br>");

    document.write(CheckIfstartsWithOne(24, 7) ? "Yes" + "</br>" : "No" + "</br>");

    document.write(CheckIfstartsWithOne(24, 15) ? "Yes" : "No");

</script>

Output :

Yes
Yes
No
Yes

Time Complexity: O(m), where m is calculated as log2(n)
Auxiliary Space: O(1), as no extra space is required
This article is contributed by Ayush Jha. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
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