Codewars Question: (Sum of Digits / Digital Root)
Given n, take the sum of the digits of n. If that value has more than one digit, continue reducing in this way until a single-digit number is produced. The input will be a non-negative integer.
Test Cases:
16 --> 1 + 6 = 7
942 --> 9 + 4 + 2 = 15 --> 1 + 5 = 6
132189 --> 1 + 3 + 2 + 1 + 8 + 9 = 24 --> 2 + 4 = 6
493193 --> 4 + 9 + 3 + 1 + 9 + 3 = 29 --> 2 + 9 = 11 --> 1 + 1 = 2
My code:
#include <bits/stdc++.h>
using namespace std;
int singleDigit(int n)
{
int ans;
while (n > 0)
{
int lastDigit = n % 10;
n /= 10;
ans += lastDigit;
}
while (ans > 9)
{
int n1 = ans;
ans = 0;
while (n1 > 0)
{
int lastDigit = n1 % 10;
n1 /= 10;
ans += lastDigit;
}
}
return ans;
}
int main()
{
cout << singleDigit(49319366) << endl;
return 0;
}
Is there a better or optimized way to solve this problem or to reduce time complexity?
This function works for non-negative integers, adapting for negative numbers is straightforward.
int singleDigit(int n)
{
return (n-1) % 9 + 1;
}
It has the following advantages:
no variables to forget to initialise
no loops to commit an off-by-one error
fast
The disadvantages are:
it is not immediately clear how or why it works
For more information on the last bullet point, see:
Direct formulas for the digital root
Modulo operation with negative numbers
I am trying to finish an assignment in C for the CS50 course in which I must implement Luhn's algorithm to validate a credit card number. Here is a quick example to elaborate:
credit card number: 4003600000000014.
Now for every other digit, starting with the number’s second-to-last digit:
1-0-0-0-0-6-0-4
Let’s multiply each of the digits by 2:
1•2 + 0•2 + 0•2 + 0•2 + 0•2 + 6•2 + 0•2 + 4•2
That gives us:
2 + 0 + 0 + 0 + 0 + 12 + 0 + 8
Now let’s add those products’ digits (i.e., not the products themselves) together:
2 + 0 + 0 + 0 + 0 + 1 + 2 + 0 + 8 = 13
Now let’s add that sum (13) to the sum of the digits that weren’t multiplied by 2 (starting from the end):
13 + 4 + 0 + 0 + 0 + 0 + 0 + 3 + 0 = 20
Yup, the last digit in that sum (20) is a 0, so the number is valid.
I figured out how to extract each number in the credit card individually (I know my way is boring and probably not practical), so the next step is to multiply every other number by two and add (the products' digits, not the digits themselves) and this is what I need help of how to do it?
MY code:
#include <cs50.h>
#include <stdio.h>
#include <math.h>
int main(void)
{
long credit_card_number;
do
{
credit_card_number = get_long("Enter your credit card number: ");
}
while (credit_card_number < 1 || credit_card_number > 9999999999999999);
//American Express uses 15-digit numbers. American Express numbers start with 34 or 37
//MasterCard uses 16-digit numbers. MasterCard numbers start with 51, 52, 53, 54, or 55.
//Visa uses 13- and 16-digit numbers. Visa numbers start with 4.
// checksum
long last_number;
long credit_card_without_last_number;
long second_to_last_number;
long credit_card_without_second_number;
long third_number;
long credit_card_without_third_number;
long fourth_number;
long credit_card_without_fourth_number;
long fifth_number;
long credit_card_without_fifth_number;
long sixth_number;
long credit_card_without_sixth_number;
long seventh_number;
long credit_card_without_seventh_number;
long eighth_number;
long credit_card_without_eighth_number;
long ninth_number;
long credit_card_without_ninth_number;
long tenth_number;
long credit_card_without_tenth_number;
long eleventh_number;
long credit_card_without_eleventh_number;
long twelfth_number;
long credit_card_without_twelfth_number;
long thirteenth_number;
long credit_card_without_thirteenth_number;
long fourteenth_number;
long credit_card_without_fourteenth_number;
long fifteenth_number;
long credit_card_without_fifteenth_number;
long sixteenth_number;
long multiply_digits;
//separating each number starting from the last (right)in its own variable.
last_number = credit_card_number % 10;
credit_card_without_last_number = credit_card_number / 10;
second_to_last_number = credit_card_without_last_number % 10;
credit_card_without_second_number = credit_card_without_last_number / 10;
third_number = credit_card_without_second_number % 10;
credit_card_without_third_number = credit_card_without_second_number / 10;
fourth_number = credit_card_without_third_number % 10;
credit_card_without_fourth_number = credit_card_without_third_number / 10;
fifth_number = credit_card_without_fourth_number % 10;
credit_card_without_fifth_number = credit_card_without_fourth_number / 10;
sixth_number = credit_card_without_fifth_number % 10;
credit_card_without_sixth_number = credit_card_without_fifth_number / 10;
seventh_number = credit_card_without_sixth_number % 10;
credit_card_without_seventh_number = credit_card_without_sixth_number / 10;
eighth_number = credit_card_without_seventh_number % 10;
credit_card_without_eighth_number = credit_card_without_seventh_number / 10;
ninth_number = credit_card_without_eighth_number % 10;
credit_card_without_ninth_number = credit_card_without_eighth_number / 10;
tenth_number = credit_card_without_ninth_number % 10;
credit_card_without_tenth_number = credit_card_without_ninth_number / 10;
eleventh_number = credit_card_without_tenth_number % 10;
credit_card_without_eleventh_number = credit_card_without_tenth_number / 10;
twelfth_number = credit_card_without_eleventh_number % 10;
credit_card_without_twelfth_number = credit_card_without_eleventh_number / 10;
thirteenth_number = credit_card_without_twelfth_number % 10;
credit_card_without_thirteenth_number = credit_card_without_twelfth_number / 10;
fourteenth_number = credit_card_without_thirteenth_number % 10;
credit_card_without_fourteenth_number = credit_card_without_thirteenth_number / 10;
fifteenth_number = credit_card_without_fourteenth_number % 10;
credit_card_without_fifteenth_number = credit_card_without_fourteenth_number / 10;
sixteenth_number = credit_card_without_fifteenth_number % 10;
//Here I need the help to multiply these numbers by two and then add each product's
//digits to the rest of the unused numbers.
multiply_digits = (second_to_last_number*2)+(fourth_number*2)+(sixth_number*2)+(eighth_number*2)+(tenth_number*2)+(twelfth_number*2)+(fourteenth_number*2)+(sixteenth_number*2);
}
Try doing this instead
int main(){
long cNo = 4003600000000014;
int arr[16];
for(int i=0; i<16; i++){
arr[15-i] = cNo % 10;
cNo /= 10;
}
int multipliedSum = 0;
for(int i=0; i<16; i++){
if(i%2==1)
multipliedSum += arr[i];
else{
if(arr[i]*2<10){
multipliedSum += (arr[i]*2);
}else{
int num = arr[i]*2;
while(num){
multipliedSum += num%10;
num/=10;
}
}
}
}
printf("valid = %s\n",multipliedSum%10==0?" True": " False");
}
You will get the following
valid = True
A general algorithm for adding digits (assuming an integer type):
Initialize your sum to 0: sum = 0
Extract the lowest digit from the number using the % modulus operator: digit = number % 10
Add the value of that digit to the sum: sum += digit (shorthand for sum = sum + digit)
Divide the number by 10: number /= 10 (shorthand for number = number / 10
If the number is non-zero after dividing by 10, go back to 2
End
The modulus operator % returns the integer remainder of an integer division - 123 / 10 == 12 rem 3. So the remainder of dividing the number by 10 is the least significant decimal digit of the number. Notice that integer division gives you an integer result - 123 / 10 == 12, not 12.3.
You'll want to put this in a separate function, so you can write something like
int sumdig( int v )
{
...
}
int main( void )
{
int value = 123;
int sum = sumdig( value ); // sumdig will return 1 + 2 + 3, or 6
...
}
When you find yourself creating a bunch of separate variables of the same type with the same name except for some tacked-on ordinal (var1, var2, var3 or first_thing, second_thing, third_thing), that's a real strong hint you want to use an array. You can use an array to store the individual digits of your card number:
int number[16];
and use the % 10 method as described above to extract the individual digits:
long tmp = credit_card_number; // use a temporary so we preserve the original card number
for ( int i = 0; i < 16; i++ )
{
number[i] = tmp % 10;
tmp /= 10;
}
This means that the least significant (rightmost) card number digit will be stored in number[0] and the most significant (leftmost) card number digit will be stored in number[15], so be aware of that. For the purposes of validating the number it doesn't matter, but if you want to display the contents of the array you'll have to take that into account.
Using an array makes it easier to extract subsets of digits:
for ( int i = 1; i < 16; i += 2 ) // hit every other element starting at element 1
{
number[i] *= 2; // multiply these digits by 2
}
That loop above executes the "1•2 + 0•2 + 0•2 + 0•2 + 0•2 + 6•2 + 0•2 + 4•2" portion of your algorithm.
You should be able to figure out the rest from there. Hope this helps.
Hint: to extract one digit from a number, mod it by 10.
So say that you want to figure out the sum of the digits of a number, say 123456, you will do the following:
(pseudocode)
number=123456;
sum=0;
loop if number is not 0{
sum+=number % 10;
number-=number % 10;
number=(int)(number/10);
}
Now try to implement it as a function, say digit(), and when you are trying to add some numbers digit-wise, say 123 and 456, just do digit(123)+digit(456) instead.
I want to write a program that adds all the numbers between 0 and 100 but my code does not add everything correctly. How do I add the next number to the number and then print the sum?
This is the code I have:
for(int i = 0; i <= 100; i++){
i+=i;
println(i);
}
The result of this shows 0, 2, 6, 14... and I need the sum of all the numbers 1 through 100.
The reason you are getting this odd result is that you add those numbers to i instead of having a dedicated collector.
int collector = 0;
for (int i = 0; i <= 100; i++) {
collector += i;
println(collector);
}
If you only want to print the sum once, move the println(collector) expression outside the loop.
There is also a mathematical formula to directly compute the sum of the first n numbers
Sum(1, n) = n * (n+1) / 2
In Processing:
int Sum(int n){
return n * (n + 1) / 2;
}
The formula works because the numbers 1 to N can be rearranged and added like this:
(1 + N) + (2 + N-1) + (3 + N-2) + . . . . + (N + N/2+1) = total
for N = 100:
(1 + 100) + (2 + 99) + (3 + 98) + . . . . + (50 + 51) = 5050
101 + 101 + 101 + . . . . + 101 = 5050
Days ago I had a job interview were they ask me how I would calculate the sum of all the numbers multiples of 2 or 5 from 1 to 10000 using a the c language. I did this:
int mult2_5()
{
int i, sum=0;
for(i = 1; i <= 10000; i++)
if(i % 2 == 0 || i % 5 == 0)
sum += i;
return sum;
}
I as wonder if it was any faster implementation that this one?
The modulus operator is inefficient. A more faster implementation would be something like this:
int multiply2_5(int max)
{
int i, x2 = 0,x5 = 0,x10 = 0;
for(i = 2; i < max; i+=2) x2 += i; // Store all multiples of 2 O(max/2)
for(i = 5; i < max; i+=5) x5 += i; // Store all multiples of 3 O(max/5)
for(i = 10; i < max; i+=10) x10 += i; // Store all multiples 10; O(max/10)
return x2+x5-x10;
}
In this solution I had to take out multiples of 10 because, 2 and 5 have 10 as multiple so on the second loop it will add multiples of 10 that already been added in the first loop; The three loops combine have O(8/10 max).
Another even better solution is if you take a mathematical approach.
You are trying to sum all numbers like this 2 + 4 + 6 + 8 ... 10000 and 5 + 10 + 15 +20 + ... 10000 this is the same of having 2 * (1 + 2 + 3 + 4 + … + 5000) and 5 * ( 1 + 2 + 3 + 4 + ... + 2000), the sum of 'n' natural number is (n * (n + 1)) (source) so you can calculate in a constant time, as it follows:
int multiply2_5(int max)
{
// x = 2 + 4 + 6 + ... = 2 * (1 + 2 + 3 +...)
// y = 5 + 10 + 15 + ... = 5 * (1 + 2 + 3 +...)
// The sun of n natural numbers is sn = (n (n + 1)) / 2
int x2 = max/ 2; // 2 * ( 1 +2 + 3 … max/2)
int x5 = max /5; // 5 * ( 1 +2 + 3 … max/5)
int x10 = max/ 10;
int sn2 = 0.5 * (x2 * (x2+1)); // (n * (n + 1)) / 2
int sn5 = 0.5 * (x5 * (x5+1));
int sn10 = 0.5 * (x10 * (x10+1));
return (2*sn2) + (5 *sn5) - (10*sn10);
}
As mentioned in an earlier answer, explicitly looping through the relevant multiples is better than testing the remainder each loop. But it is not necessary to calculate the multiples of 10 and subtract. Just start at 5 and step by 10 to skip them all together.
int multiply2_5b(int max)
{
int i, x2 = 0,x5 = 0;
for (i = 2; i < max; i += 2) x2 += i; // Sum all multiples of 2
for (i = 5; i < max; i += 10) x5 += i; // Sum all odd multiples of 5
return x2 + x5;
}
Just work it out on paper first, if that's what you mean by "faster".
$2\sum_{1<=2k<=10000}k + 5\sum_{1<=5k<=10000} - 10\sum_{1<=10k<=10000}k$
Sorry, my SO equation-fu is weak...Anyway this route will give you something you can almost handle on paper: 5000*6001 after reducing a few steps
int
mult2_5(void)
{
return 5000*6001;
}
Project Euler problem 1 is very similar. There's lots of folks who've posted their solution to this one.
This can just be done using math. Something like 2 * sum(1 to 5000) + 5 * sum(1 to 2000) - 10 * sum(1 to 1000). off-by-one errors left as exercise.
I almost got to a pure and simple multiplication by doing a simple loop that starts with 35 (sum of 2 + 4 + 5 + 6 + 8 + 10) with a step of 60, as that's how much your result will increase by when you take the next lot e.g. 12 + 14 + 15 + 16 + 18 + 20 etc. for(int i=35;i<5976;i=i+60) { sum=sum+i }
The 5976 comes from 5975 being the last row of numbers that end in 1000 i.e. 992 + 994 + 995 + 996 + 998 + 1000.
So it turns out that this loop runs 100 times, increasing the sum by 35 the first turn and increasing by 60 the remaining 99 times. Which is now reducable to a simple multiplication or such.
I have had to start to learning C as part of a project that I am doing. I have started doing the 'euler' problems in it and am having trouble with the first one. I have to find the sum of all multiples of 3 or 5 below 1000. Could someone please help me. Thanks.
#include<stdio.h>
int start;
int sum;
int main() {
while (start < 1001) {
if (start % 3 == 0) {
sum = sum + start;
start += 1;
} else {
start += 1;
}
if (start % 5 == 0) {
sum = sum + start;
start += 1;
} else {
start += 1;
}
printf("%d\n", sum);
}
return(0);
}
You've gotten some great answers so far, mainly suggesting something like:
#include <stdio.h>
int main(int argc, char * argv[])
{
int i;
int soln = 0;
for (i = 1; i < 1000; i++)
{
if ((i % 3 == 0) || (i % 5 == 0))
{
soln += i;
}
}
printf("%d\n", soln);
return 0;
}
So I'm going to take a different tack. I know you're doing this to learn C, so this may be a bit of a tangent.
Really, you're making the computer work too hard for this :). If we figured some things out ahead of time, it could make the task easier.
Well, how many multiples of 3 are less than 1000? There's one for each time that 3 goes into 1000 - 1.
mult3 = ⌊ (1000 - 1) / 3 ⌋ = 333
(the ⌊ and ⌋ mean that this is floor division, or, in programming terms, integer division, where the remainder is dropped).
And how many multiples of 5 are less than 1000?
mult5 = ⌊ (1000 - 1) / 5 ⌋ = 199
Now what is the sum of all the multiples of 3 less than 1000?
sum3 = 3 + 6 + 9 + ... + 996 + 999 = 3×(1 + 2 + 3 + ... + 332 + 333) = 3×∑i=1 to mult3 i
And the sum of all the multiples of 5 less than 1000?
sum5 = 5 + 10 + 15 + ... + 990 + 995 = 5×(1 + 2 + 3 + ... + 198 + 199) = 5×∑i = 1 to mult5 i
Some multiples of 3 are also multiples of 5. Those are the multiples of 15.
Since those count towards mult3 and mult5 (and therefore sum3 and sum5) we need to know mult15 and sum15 to avoid counting them twice.
mult15 = ⌊ (1000 - 1) /15 ⌋ = 66
sum15 = 15 + 30 + 45 + ... + 975 + 990 = 15×(1 + 2 + 3 + ... + 65 + 66) = 15×∑i = 1 to mult15 i
So the solution to the problem "find the sum of all the multiples of 3 or 5 below 1000" is then
soln = sum3 + sum5 - sum15
So, if we wanted to, we could implement this directly:
#include <stdio.h>
int main(int argc, char * argv[])
{
int i;
int const mult3 = (1000 - 1) / 3;
int const mult5 = (1000 - 1) / 5;
int const mult15 = (1000 - 1) / 15;
int sum3 = 0;
int sum5 = 0;
int sum15 = 0;
int soln;
for (i = 1; i <= mult3; i++) { sum3 += 3*i; }
for (i = 1; i <= mult5; i++) { sum5 += 5*i; }
for (i = 1; i <= mult15; i++) { sum15 += 15*i; }
soln = sum3 + sum5 - sum15;
printf("%d\n", soln);
return 0;
}
But we can do better. For calculating individual sums, we have Gauss's identity which says the sum from 1 to n (aka ∑i = 1 to n i) is n×(n+1)/2, so:
sum3 = 3×mult3×(mult3+1) / 2
sum5 = 5×mult5×(mult5+1) / 2
sum15 = 15×mult15×(mult15+1) / 2
(Note that we can use normal division or integer division here - it doesn't matter since one of n or n+1 must be divisible by 2)
Now this is kind of neat, since it means we can find the solution without using a loop:
#include <stdio.h>
int main(int argc, char *argv[])
{
int const mult3 = (1000 - 1) / 3;
int const mult5 = (1000 - 1) / 5;
int const mult15 = (1000 - 1) / 15;
int const sum3 = (3 * mult3 * (mult3 + 1)) / 2;
int const sum5 = (5 * mult5 * (mult5 + 1)) / 2;
int const sum15 = (15 * mult15 * (mult15 + 1)) / 2;
int const soln = sum3 + sum5 - sum15;
printf("%d\n", soln);
return 0;
}
Of course, since we've gone this far we could crank out the entire thing by hand:
sum3 = 3×333×(333+1) / 2 = 999×334 / 2 = 999×117 = 117000 - 117 = 116883
sum5 = 5×199×(199+1) / 2 = 995×200 / 2 = 995×100 = 99500
sum15 = 15×66×(66+1) / 2 = 990×67 / 2 = 495 × 67 = 33165
soln = 116883 + 99500 - 33165 = 233168
And write a much simpler program:
#include <stdio.h>
int main(int argc, char *argv[])
{
printf("233168\n");
return 0;
}
You could change your ifs:
if ((start % 3 == 0) || (start % 5 == 0))
sum += start;
start ++;
and don´t forget to initialize your sum with zero and start with one.
Also, change the while condition to < 1000.
You would be much better served by a for loop, and combining your conditionals.
Not tested:
int main()
{
int x;
int sum = 0;
for (x = 1; x <= 1000; x++)
if (x % 3 == 0 || x % 5 == 0)
sum += x;
printf("%d\n", sum);
return 0;
}
The answers are all good, but won't help you learn C.
What you really need to understand is how to find your own errors. A debugger could help you, and the most powerful debugger in C is called "printf". You want to know what your program is doing, and your program is not a "black box".
Your program already prints the sum, it's probably wrong, and you want to know why. For example:
printf("sum:%d start:%d\n", sum, start);
instead of
printf("%d\n", sum);
and save it into a text file, then try to understand what's going wrong.
does the count start with 1 and end with 999?
does it really go from 1 to 999 without skipping numbers?
does it work on a smaller range?
Eh right, well i can see roughly where you are going, I'm thinking the only thing wrong with it has been previously mentioned. I did this problem before on there, obviously you need to step through every multiple of 3 and 5 and sum them. I did it this way and it does work:
int accumulator = 0;
int i;
for (i = 0; i < 1000; i += 3)
accumulator += i;
for (i = 0; i < 1000; i +=5) {
if (!(i%3==0)) {
accumulator += i;
}
}
printf("%d", accumulator);
EDIT: Also note its not 0 to 1000 inclusive, < 1000 stops at 999 since it is the last number below 1000, you have countered that by < 1001 which means you go all the way to 1000 which is a multiple of 5 meaning your answer will be 1000 higher than it should be.
You haven't said what the program is supposed to do, or what your problem is. That makes it hard to offer help.
At a guess, you really ought to initialize start and sum to zero, and perhaps the printf should be outside the loop.
Really you need a debugger, and to single-step through the code so that you can see what it's actually doing. Your basic problem is that the flow of control isn't going where you think it is, and rather than provide correct code as others have done, I'll try to explain what your code does. Here's what happens, step-by-step (I've numbered the lines):
1: while (start < 1001) {
2: if (start % 3 == 0) {
3: sum = sum + start;
4: start += 1;
5: }
6: else {
7: start += 1;
8: }
9:
10: if (start % 5 == 0) {
11: sum = sum + start;
12: start += 1;
13: }
14: else {
15: start += 1;
16: }
17: printf("%d\n", sum);
18: }
line 1. sum is 0, start is 0. Loop condition true.
line 2. sum is 0, start is 0. If condition true.
line 3. sum is 0, start is 0. sum <- 0.
line 4. sum is 0, start is 0. start <- 1.
line 5. sum is 0, start is 1. jump over "else" clause
line 10. sum is 0, start is 1. If condition false, jump into "else" clause.
line 15. sum is 0, start is 1. start <- 2.
line 16 (skipped)
line 17. sum is 0, start is 2. Print "0\n".
line 18. sum is 0, start is 2. Jump to the top of the loop.
line 1. sum is 0, start is 2. Loop condition true.
line 2. sum is 0, start is 2. If condtion false, jump into "else" clause.
line 7. sum is 0, start is 2. start <- 3.
line 10. sum is 0, start is 3. If condition false, jump into "else" clause.
line 15. sum is 0, start is 3. start <- 4.
line 17. sum is 0, start is 4. Print "0\n".
You see how this is going? You seem to think that at line 4, after doing sum += 1, control goes back to the top of the loop. It doesn't, it goes to the next thing after the "if/else" construct.
You have forgotten to initialize your variables,
The problem with your code is that your incrementing the 'start' variable twice. This is due to having two if..else statements. What you need is an if..else if..else statement as so:
if (start % 3 == 0) {
sum = sum + start;
start += 1;
}
else if (start % 5 == 0) {
sum = sum + start;
start += 1;
}
else {
start += 1;
}
Or you could be more concise and write it as follows:
if(start % 3 == 0)
sum += start;
else if(start % 5 == 0)
sum += start;
start++;
Either of those two ways should work for you.
Good luck!
Here's a general solution which works with an arbitrary number of factors:
#include <stdio.h>
#define sum_multiples(BOUND, ...) \
_sum_multiples(BOUND, (unsigned []){ __VA_ARGS__, 0 })
static inline unsigned sum_single(unsigned bound, unsigned base)
{
unsigned n = bound / base;
return base * (n * (n + 1)) / 2;
}
unsigned _sum_multiples(unsigned bound, unsigned bases[])
{
unsigned sum = 0;
for(unsigned i = 0; bases[i]; ++i)
{
sum += sum_single(bound, bases[i]);
for(unsigned j = i + 1; bases[j]; ++j)
sum -= sum_single(bound, bases[i] * bases[j]);
}
return sum;
}
int main(void)
{
printf("%u\n", sum_multiples(999, 3, 5));
return 0;
}