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
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