Problem
Consider the sequence D of the last decimal digits of the first N Fibonacci numbers, i.e. D = (F0%10,F1%10,…,FN−1%10).
Now, you should perform the following process:
Let D=(D1,D2,…,Dl)
If l=1, the process ends.
Create a new sequence
E=(D2,D4,…,D2⌊l/2⌋)
In other words, E is the sequence created by removing all odd-indexed elements from D
Change D to E
When this process terminates, the sequence D
contains only one number. You have to find this number.
Input
The first line of the input contains a single integer T
denoting the number of test cases.
The description of T test cases follows.
The first and only line of each test case contains a single integer N
Output
For each test case, print a single line containing one integer ― the last remaining number.
Code
#include <stdio.h>
#include <stdlib.h>
int test(int *arr, int n);
int main() {
int t;
scanf("%d", &t);
while (t--) {
int n;
scanf("%d", &n);
int *arr;
arr = (int *)malloc((n + 1) * sizeof(int));
arr[1] = 0;
arr[2] = 1;
for (int i = 3; i <= n; i++) {
arr[i] = arr[i - 1] + arr[i - 2];
}
/*
for(int k=1;k<=n;k++){
printf("%d ",arr[k] );
}
printf("\n");
*/
printf("%d\n", (test(arr, n)) % 10);
}
}
int test(int *arr, int n) {
if (n == 1) {
return arr[1];
} else {
for (int i = 1; i <= (n / 2); i++) {
arr[i] = arr[2 * i];
}
return test(arr, n / 2);
}
}
Using the algorithm from https://math.stackexchange.com/questions/681674/recursively-deleting-every-second-element-in-a-list,
Find the largest integer A, such that 2^A < N.
Find Fibonnaci(2^A - 1) % 10
Adding to Bill Lynch's answer, which is itself based on this other answer by happymath:
You will always end up getting 2n − 1 where n is maximum integer such that 2n < K
I'd like to point out another useful mathematical property.
In number theory, the nth Pisano period, written π(n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats.
(https://en.wikipedia.org/wiki/Pisano_period)
Here we need to consider the case where n = 10, π(10) = 60 and the last decimal digits correspond to the OEIS sequence A003893:
0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1
So that there's no need to calculate the actual Fibonacci number, nor to generate all the sequence up to N.
I want to do this in a function: How do I find out in a C program if a number is divisible by 2, 3, 4, 5, 6, 8, 9, 25 and 125 without using the % operator and using the divisibility rules? the base should be 10*
To use divisibility rules, you have to work with digits. Perhaps task assumes no division (at least in explicit form - you can extract digits from string representation)
For divisibility by 2, check whether the last digit is in set 0,2,4,6,8
For divisibility by 4, check whether the last digit + doubled previous one is in set 0,4,8. If result is larger than 10, repeat (88=>2*8+8=24=>2*2+4=8)
Similar situation for 8, but sum last + 2*previous + 4*second_from_the_end (512 => 4*5+2*1+2=24=>2*2+4=8)
For divisibility by 5, check whether the last digit is in set 0,5, similar situation for 25, 125
For divisibility by 3, sum all digits, repeat process until result becomes < 10. So-called "digit root" should be in set 0,3,6,9, similar situation for divisibility by 9.
For 6 check divisibilty by both 2 and by 3
I am not strong in C, so my example perhaps is very weird (ideone check)
#include <stdio.h>
int divby3(int n) {
char s[10];
do {
sprintf(s, "%d", n); //transform 72 to char buffer "72"
n = 0;
int i = 0;
while(s[i]) //until nil (end of string) found, we can also use for loop
n += s[i++] - 0x30; //get difference of current char and char "0"
}
while (n >= 10); //until 1-digit value
return (n==0) || (n==3) || (n==6) || (n==9);
}
int divby5(int n) {
char s[10];
int len = sprintf(s, "%d", n);
n = s[len - 1] - 0x30; //last digit
return (n==0) || (n==5);
}
int main(void) {
printf("%d", divby3(72)); //try 71
return 0;
}
A function that uses the a - (a / b * b) implementation of the modulus operator: (credit #MikeCat)
bool isDivisible(int a, int b)
{
if((a - (a / b * b)) == 0) return true;
return false;
}
Usage:
int main(void)
{
int a = 8;
int b = 4;
int c = 3;
bool res = isDivisible(a,b);
bool res2 = isDivisible(a,c);
return 0;
}
EDIT - to address question in comments:
"how can i represent such a program with the divisibility rules? Thank you for your code, i forgott to mention that i have to use the divisibility rules in each function"
The following shows how to pass in divisibility rules as an argument...
const int div_1[] = {2, 3, 4, 5, 6, 8, 9, 25, 125};
const int div_2[] = {7, 5, 17, 12, 11};
int main()
{
size_t size = 0;
size = sizeof div_1/sizeof div_1[0];
bool res = isDivisible(2*3*4*5*6*8*9*25*125, div_1, size);
size = sizeof div_2/sizeof div_2[0];
bool res2 = isDivisible(125, div_2, size);
return 0;
}
// numerator divisor array array size
bool isDivisible(long a, long div_rules[], size_t size)
{
//divisibility rules
const int divisors[] = {2, 3, 4, 5, 6, 8, 9, 25, 125};
for(int i = 0; i<size;i++)
{
if((a - (a / div_rules[i] * div_rules[i])) != 0) return false;
}
return true;
}
I am trying to make a program that will count the number of even numbers in the provided arrays. When I run the program now, it will return the amount of numbers in the array, but not the amount of even numbers. For some reason my count_even function doesn't work. Can anyone help?
#include <stdio.h>
int main()
{
int data_array_1[] = { 1, 3, 5, 7, 9, 11 };
int data_array_2[] = { 2, -4, 6, -8, 10, -12, 14, -16 };
int data_array_3[] = { 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 };
int data_array_4[] = { 6, 2, 4, 5, 1, -9 };
int data_array_5[] = { 1, 3, 9, 23, 5, -2, 4 };
int result_1 = count_even(data_array_1, 6);
printf("data_array_1 has %d even numbers.\n", result_1);
int result_2 = count_even(data_array_2, 8);
printf("data_array_2 has %d even numbers.\n", result_2);
int result_3 = count_even(data_array_3, 11);
printf("data_array_3 has %d even numbers.\n", result_3);
int result_4 = count_even(data_array_4, 6);
printf("data_array_4 has %d even numbers.\n", result_4);
int result_5 = count_even(data_array_5, 7);
printf("data_array_5 has %d even numbers.\n", result_5);
return 0;
}
int count_even(int* data_array, int size)
{
int even_num = 0;
for (int i = 0; i == size; i++)
{
if (data_array[size] % 2 == 0)
{
even_num++;
}
}
return even_num;
}
The condition in your for loop is wrong.
The correct condition should say "as long as the index is smaller than size", but yours say "as long as the index equal to to size".
The condition should be i < size.
As for the result, it seems like it should return 0 (for the non-working code), not size.
Also, you are using size as an index, when you should use i.
In your count_even function, you are using the size attribute as the array index, when it should be i
int count_even(int* data_array, int size)
{
int even_num = 0
for(int i = 0; i <= size, ++i)
{
if(data_array[i] % 2 == 0)
{
even_num++;
}
}
return even_num;
}
these two lines are the root of the problems in the code:
for (int i = 0; i == size; i++)
{
if (data_array[size] % 2 == 0)
the for() statement, should be:
for (int i = 0; i < size; i++)
so the loop exits when reaching the end of the array
the if() statement is always looking at the same entry beyond the end of the array, This is undefined behaviour
The if() statement should be:
if (data_array[i] % 2 == 0)
However, the modulo operator & is not a good choice for negative numbers
a better choice would be:
if ( !(data_array[i] & 1 ) )
I have a C array fftArray[64] that contains values that I want averaged and placed into another array frequencyBar[8]. Getting the average of the entire array would be easy enough using a for statement.
int average, sum = 0;
for (i = 0; i < 64; i++)
{
sum += fftArray[i];
}
average = sum/64;
But I just can't seem to figure out how to get the average from fftArray[0] through fftArray[8] and store this in frequencyBar[0], the average of fftArray[9] through fftArray[16] and store this in frequencyBar[1], etc. Can anyone help me out with this? Thanks
This looks like a homework assignment, so, rather than give you the outright answer, I'd rather just point you in the right direction...
use a nested loop (one inside the other). One loop cycles 0-7, the other one 0 - 63. Use the smaller one to populate your sliced averages.
or better yet use the % operator to see when you've gone through 8 elements and do an average of your total, then reset the total for the next set. Then you'll have learned how to use the % operator too! :)
[EDIT]
ok, if not homework then something like this... I haven't written C in 5 years, so treat this as pseudo code:
//assuming you have a fftArray[64] with data, as per your question
int i,sum,avCounter,total;
int averages[8];
for(i=0 , avCounter=0, total=0 ; i<64; ){
total += fftArray[i];
if(++i % 8 == 0){ //%gives you the remainder which will be 0 every 8th cycle
averages[avCounter++] = total / 8
total = 0; //reset for next cycle
}
}
I think this will work better than a nested loop... but I'm not sure since % is division which is more processor heavy than addition... however... I doubt anyone would notice :)
int i, j;
for (i = 0; i < 8; i++) {
int sum = 0;
for (j = 0; j < 8; j++) {
sum += fftArray[ 8*i + j ];
}
frequencyBar[i] = sum / 8;
}
Bonus exercise: Optimize this code for speed on your chosen platform.
TF,
DISCLAIMER: This code is just off the top of my head... it hasn't even been compiled, let alone tested.
// returns the average of array[first..last] inclusive.
int average(int[] array, int first, int last) {
int sum = 0;
for (i = first; i <= last; i++)
sum += array[i];
return sum / (last - first + 1); // not sure about the +1
}
Then what you'd do is loop through the indexes of your frequencyBar array [0..7], setting frequencyBar[i] = average(array, first, last);... the tricky bit is calculating the first and last indexes... try i*8 and (i+1)*8 respectively... that may not be exactly right, but it'll be close ;-)
Cheers. Keith.
EDIT: Bored... waiting for my test results to come back. No news is good news, right? ;-)
It turns out that passing the length is a fair bit simpler than passing the last index.
#include <stdio.h>
int sum(int array[], int first, int length) {
int sum = 0;
for (int i = first; i < first+length; i++)
sum += array[i];
return sum;
}
double average(int array[], int first, int length) {
double total = sum(array, first, length);
#ifdef DEBUG
printf("DEBUG: [%2d..%2d] %d", first, first+length-1, array[first]);
for (int i = first+1; i < first+length; i++)
printf(" + %d", array[i]);
printf(" = %d / %d = %f\n", (int)total, length, total/length);
#endif
return total / length;
}
int main(int argc, char* argv[]) {
int array[] = { // average
1, 2, 3, 4, 5, 1, 2, 3, // 2.625
4, 5, 1, 2, 3, 4, 5, 1, // 3.125
2, 3, 4, 5, 1, 2, 3, 4, // 3
5, 1, 2, 3, 4, 5, 1, 2, // 2.875
3, 4, 5, 1, 2, 3, 4, 5, // 3.375
1, 2, 3, 4, 5, 1, 2, 3, // 2.625
4, 5, 1, 2, 3, 4, 5, 1, // 3.125
2, 3, 4, 5, 1, 2, 3, 4 // 3
};
double frequency[8];
for (int i = 0; i < 8; i++)
frequency[i] = average(array, i*8, 8);
for (int i = 0; i < 8; i++)
printf("%f ", frequency[i]);
printf("\n");
}
Watch your sum doesn't wrap around if fftArray has large value in!
Using a microcontroller (PIC18F4580), I need to collect data and send it to an SD card for later analysis. The data it collects will have values between 0 and 1023, or 0x0 and 0x3FF.
So what I need to do is convert 1023 into a base 10 string of literal ASCII values (0x31, 0x30, 0x32, 0x33, ...).
My problem is that the only way I can think of to split the digits apart requires a lot of division.
char temp[4];
temp[0] = 1023 % 10;
temp[1] = (1023 % 100) / 10;
temp[2] = (1023 % 1000) / 100;
temp[3] = (1023 % 10000) / 1000;
Using this method, finding the ASCII values of an n digit decimal number requires 2n-1 divisions. Is there a method that would be faster?
The end goal of this is to wind up with a .csv file on the SD card that can quickly be plugged into any laptop to see a graph of the data in Excel.
The obvious solution is not to convert the data to ASCII at all but store it in binary format. That way all you need to worry about is the endianness of the data. If the system performing the later analysis is far more powerful than your embedded target, then it would make sense to let that deal with the conversion and and byte order.
On the other hand, it is possible that the execution time of the / and % is insignificant compared to the time taken to transfer the data to the SD card; so make sure that you are optimising the right thing.
There's certainly a much faster way: have an array of 1024 pre-computed strings. Then you can just do bounds checking, followed by an index into the array.
It's unclear from your question whether your code is running on the microcontroller though. If that's the case, you may not have enough memory for this approach.
I agree with what Clifford said, that you shouldn't worry about optimizing it if you don't have to, and that you can push the log cleanup to your analysis platform, rather than worrying about formatting in an embedded application.
That being said, here's an article that might be useful to you. It uses a loop, shifts, additions and branches, with linear/constant complexity: http://www.johnloomis.org/ece314/notes/devices/binary_to_BCD/bin_to_bcd.html
Also, I thought it would be fun to make some code that doesn't perform any divides, multiplies, or branches, but still gives the correct answer [0 - 1024). No promises that this is any faster than other options. This sort of code is just an option to explore.
I'd love to see if anyone can provide some tricks to make the code smaller, require less memory, or require fewer operations, while keeping the rest of the counts equal, or shrinking them :)
Stats:
224 bytes in constants (no idea on the code size)
5 bit-shift-rights
3 subtracts
5 bitwise-ands
4 bitwise-ors
1 greater-than comparison
Perf:
Using the perf comparisons and itoa routines in Jonathan Leffler's answer, here are the stats I got:
Division 2.15
Subtraction 4.87
My solution 1.56
Brute force lookup 0.36
I increased the iteration count to 200000 to ensure I didn't have any problems with timing resolution, and had to add volatile to the function signatures so that the compiler didn't optimize out the loop. I used VS2010 express w/ vanilla "release" settings, on a 3ghz dual core 64 bit Windows 7 machine (tho it compiled to 32 bit).
The code:
#include "stdlib.h"
#include "stdio.h"
#include "assert.h"
void itoa_ten_bits(int n, char s[])
{
static const short thousands_digit_subtract_map[2] =
{
0, 1000,
};
static const char hundreds_digit_map[128] =
{
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9,
0, 0, 0,
};
static const short hundreds_digit_subtract_map[10] =
{
0, 100, 200, 300, 400, 500, 600, 700, 800, 900,
};
static const char tens_digit_map[12] =
{
0, 1, 2, 3, 3, 4, 5, 6, 7, 7, 8, 9,
};
static const char ones_digit_map[44] =
{
0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
0, 1, 2, 3
};
/* Compiler should optimize out appX constants, % operations, and + operations */
/* If not, use this:
static const char ones_digit_append_map[16] =
{
0, 6, 2, 8, 4, 10, 6, 12, 8, 14, 10, 16, 12, 18, 14, 20,
};
*/
static const char a1 = 0x10 % 10, a2 = 0x20 % 10, a3 = 0x40 % 10, a4 = 0x80 % 10;
static const char ones_digit_append_map[16] =
{
0, a1, a2, a1 + a2,
a3, a1 + a3, a2 + a3, a1 + a2 + a3,
a4, a1 + a4, a2 + a4, a1 + a2 + a4,
a3 + a4, a1 + a3 + a4, a2 + a3 + a4, a1 + a2 + a3 + a4,
};
char thousands_digit, hundreds_digit, tens_digit, ones_digit;
assert(n >= 0 && n < 1024 && "n must be between [0, 1024)");
/* n &= 0x3ff; can use this instead of the assert */
thousands_digit = (n >> 3 & 0x7f) > 0x7c;
n -= thousands_digit_subtract_map[thousands_digit];
ones_digit = ones_digit_map[
(n & 0xf)
+ ones_digit_append_map[n >> 4 & 0xf]
+ ones_digit_append_map[n >> 8 & 0x3]
];
n -= ones_digit;
hundreds_digit = hundreds_digit_map[n >> 3 & 0x7f];
n -= hundreds_digit_subtract_map[hundreds_digit];
tens_digit = tens_digit_map[n >> 3];
s[0] = '0' | thousands_digit;
s[1] = '0' | hundreds_digit;
s[2] = '0' | tens_digit;
s[3] = '0' | ones_digit;
s[4] = '\0';
}
int main(int argc, char* argv)
{
int i;
for(i = 0; i < 1024; ++i)
{
char blah[5];
itoa_ten_bits(i, blah);
if(atoi(blah) != i)
printf("failed %d %s\n", i, blah);
}
}
There's a way of doing it using subtractions, but I am not convinced it's faster than using subtractions and modulus on a "normal" CPU (may be different in an embedded environment).
Something like this:
char makedigit (int *number, int base)
{
static char map[] = "0123456789";
int ix;
for (ix=0; *number >= base; ix++) { *number -= base; }
return map[ix];
}
char *makestring (int number)
{
static char tmp[5];
tmp[0] = makedigit(&number, 1000);
tmp[1] = makedigit(&number, 100);
tmp[2] = makedigit(&number, 10);
tmp[3] = makedigit(&number, 1);
tmp[4] = '\0';
return tmp;
}
Then, a call to makestring() should result in a (static, so copy it before overwriting) string with the converted number (zero-prefixed, at 4 characters width, as the original assumption is a value in the 0-1023 range).
With some care in finding the right number(s) to use, you can multiply by the reciprocal of the base rather than dividing by the base. Terje's code is for an x86, but porting the general idea to a PIC shouldn't be tremendously difficult.
If the values are correctly in range (0..1023), then your last conversion is unnecessarily wasteful on the divisions; the last line could be replaced with:
temp[3] = 1023 / 1000;
or even:
temp[3] = 1023 >= 1000;
Since division is repeated subtraction, but you have a very special case (not a general case) division to deal with, I'd be tempted to compare the timings for the following code with the division version. I note that you put the digits into the string in 'reverse order' - the least significant digit goes in temp[0] and the most in temp[4]. Also, there is no chance of null-terminating the string given the storage. This code uses a table of 8 bytes of static data - considerably less than many of the other solutions.
void convert_to_ascii(int value, char *temp)
{
static const short subtractors[] = { 1000, 100, 10, 1 };
int i;
for (i = 0; i < 4; i++)
{
int n = 0;
while (value >= subtractors[i])
{
n++;
value -= subtractors[i];
}
temp[3-i] = n + '0';
}
}
Performance testing - Intel x86_64 Core 2 Duo 3.06 GHz (MacOS X 10.6.4)
This platform is probably not representative of your microcontroller, but the test shows that on this platform, the subtraction is considerably slower than the division.
void convert_by_division(int value, char *temp)
{
temp[0] = (value % 10) + '0';
temp[1] = (value % 100) / 10 + '0';
temp[2] = (value % 1000) / 100 + '0';
temp[3] = (value % 10000) / 1000 + '0';
}
void convert_by_subtraction(int value, char *temp)
{
static const short subtractors[] = { 1000, 100, 10, 1 };
int i;
for (i = 0; i < 4; i++)
{
int n = 0;
while (value >= subtractors[i])
{
n++;
value -= subtractors[i];
}
temp[3-i] = n + '0';
}
}
#include <stdio.h>
#include <timer.h>
#include <string.h>
static void time_convertor(const char *tag, void (*function)(void))
{
int r;
Clock ck;
char buffer[32];
clk_init(&ck);
clk_start(&ck);
for (r = 0; r < 10000; r++)
(*function)();
clk_stop(&ck);
printf("%s: %12s\n", tag, clk_elapsed_us(&ck, buffer, sizeof(buffer)));
}
static void using_subtraction(void)
{
int i;
for (i = 0; i < 1024; i++)
{
char temp1[4];
convert_by_subtraction(i, temp1);
}
}
static void using_division(void)
{
int i;
for (i = 0; i < 1024; i++)
{
char temp1[4];
convert_by_division(i, temp1);
}
}
int main()
{
int i;
for (i = 0; i < 1024; i++)
{
char temp1[4];
char temp2[4];
convert_by_subtraction(i, temp1);
convert_by_division(i, temp2);
if (memcmp(temp1, temp2, 4) != 0)
printf("!!DIFFERENCE!! ");
printf("%4d: %.4s %.4s\n", i, temp1, temp2);
}
time_convertor("Using division ", using_division);
time_convertor("Using subtraction", using_subtraction);
time_convertor("Using division ", using_division);
time_convertor("Using subtraction", using_subtraction);
time_convertor("Using division ", using_division);
time_convertor("Using subtraction", using_subtraction);
time_convertor("Using division ", using_division);
time_convertor("Using subtraction", using_subtraction);
return 0;
}
Compiling with GCC 4.5.1, and working in 32-bit, the average timings were (optimization '-O'):
0.13 seconds using division
0.65 seconds using subtraction
Compiling and working in 64-bit, the average timings were:
0.13 seconds using division
0.48 seconds using subtraction
Clearly, on this machine, using subtraction is not a winning proposition. You would have to measure on your machine to make a decision. And removing the modulo 10000 operation will only skew results in favour of the division (it knocks about 0.02 seconds off the time with division when replaced with the comparison; that's a 15% saving and worth having).
Is there some reason that you're particularly concerned about this?
If your compiler and C library provide an itoa() function, use that, and then worry about writing this code (and associated tests and so forth to make sure you got it right!) if for some reason that turns out to be too slow or doesn't fit into RAM or something.
I've replaced my previous answer with a better one. This code creates a 4-character string in the proper order, most significant digit in output[0] to least significant in output[3] with a zero terminator in output[4]. I don't know anything about your PIC controller or C compiler, but this code doesn't require anything more than 16-bit integers, addition/subtraction, and shifting.
int x;
char output[5];
output[4] = 0;
x = 1023;
output[3] = '0' + DivideByTenReturnRemainder(&x);
output[2] = '0' + DivideByTenReturnRemainder(&x);
output[1] = '0' + DivideByTenReturnRemainder(&x);
output[0] = '0' + x;
The key to this is the magical function DivideByTenReturnRemainder. Without using division explicitly it's still possible to divide by powers of 2 by shifting right; the problem is that 10 isn't a power of 2. I've sidestepped that problem by multiplying the value by 25.625 before dividing by 256, letting integer truncation round down to the proper value. Why 25.625? Because it's easily represented by powers of 2. 25.625 = 16 + 8 + 1 + 1/2 + 1/8. Again, multiplying by 1/2 is the same as shifting right one bit, and multiplying by 1/8 is shifting right by 3 bits. To get the remainder, multiply the result by 10 (8+2) and subtract it from the original value.
int DivideByTenReturnRemainder(int * p)
{
/* This code has been tested for an input range of 0 to 1023. */
int x;
x = *p;
*p = ((x << 4) + (x << 3) + x + (x >> 1) + (x >> 3)) >> 8;
return x - ((*p << 3) + (*p << 1));
}
Are you required to use an ASCII string of the decimal representation? It would be much easier to store it in hexadecimal format. No division required, only (relatively cheap) shift operations. Excel should be able to read it if you prepend a '0x' to each number.