Question

Number Theory Problem: To transmit a positive integer less than 1000, the Networked Number Node offers two options.

Option 1. Pay $d to send each digit d. Therefore, 987 would cost $9 + $8 + $7 = $24 to transmit.
Option 2. Encode integer into binary (base 2) first, and then pay $d to send each digit d. Therefore, 987 becomes 1111011011 and would cost $1 + $1 + $1 + $1 + $0 + $1 + $1 + $0 + $1 + $1 = $8.
What is the largest integer less than 1000 that costs the same whether using Option 1 or Option 2?

Answer 1

Answer: 503

Explanation:

First, we need to find the largest possible value when sending a number with Option 2. If we had 10 1s the smallest binary number would be:

This is greater than 1000, so the greatest possible cost when sending with option 2 will be 9. We can look at the largest numbers less than 1000 which cost 9 with Option 1 and see if they cost 9 with option 2. The largest numbers are:

900, 810, 801, 720, 711, 702,...

The smallest possible number with 10 digits and cost 9 in Option 2 is: 10111111112 = 767

Below this, we would have:

which doesn't work. We can quickly check the numbers above and see that they cost less than 9 with method 2. So, we now need to consider numbers with cost of 8. The largest numbers with a cost of 8 in Option 1 are:

It is possible to check these in base 2 and see which is the first to cost 8 with Option 2, or we can go the other way and look at numbers with a cost of 8 in Option 2. Either way, we will find the largest possible integer with a cost of 8 is:

111110111₂ = 503

We must check and make sure that there are no numbers larger than 503 with an Option 2 cost lower than 8. The numbers with cost 7 in Option 1 with value greater than 503 are 700, 610, 601, and 520. We can check that all cost less than 7 in Option 2 and can be eliminated. The numbers with cost 6 in Option 1 with value greater than 503 are 600 and 510, neither of which have cost 6 in Option 2 and therefore do not work. Since a number with cost 5 or lower must be less than 500, the largest possible integer is 503

Answer 2

In C, int requires 4 butes to sotre a integer number. that means it requires 4*8 = 32 bits. The maximum number can be stored using 32 bits is

The first digit is used for sign. decimal equivalent to this number can be computed as

=

=

That means int can store a up to 2147483646.

Testing with C code

#include<stdio.h>

int main()

{

int a = 2147483647; // a with max number

printf("%d\\",a);// printing max number

a = a+1;// adding one to the number

printf("%d\\",a);// printing the number after adding one

return 0;

}

THE OUTPUT

$ ./a.out

2147483647

-2147483648

Explanation:

PLEASE LIKE THIS OR LET IT BE HELPFUl