How would you go about performing a modulo operation on a fairly large number where modular exponentiation approaches cannot be taken?
For example, take this prime number modulo operation:
6864797660130609714981900799081393217269435300143305409394463459185543183
3976560521225596406614545549772963113914808580371219879997166438125740282
91115057151 % 4
WolframAlpha tells me it's 3. That's great, but I'd like to write an algorithm so that my own calculator app can handle that.
I'm assuming that for such a large number I would store the number in an array, one element per digit.
I don't have enough reputation to comment, but these two guys that say, that you can look only at last digits are wrong, take %7 for example - you always have to look at all digits.
As you probably know (a+b)%n = (a%n + b%n) % n and (a*b)%n = (a%n * b%n) % n Using that we can first calculate 1%n, 10%n, 100%n and so on, then multiply these values by digits you have in your number and at the end add it all together.
I wrote it in c++:
//assume we have number of length len in reversed order
//example: 123%9 -> n = 9, num[0] = 3, num[1] = 2; num[2] = 1, len = 3
int mod(int n, int num[], int len)
{
int powersOf10modn = 1;
int anwser = 0;
for(int i = 0; i < len; i++)
{
anwser = (anwser + powersOf10modn * num[i]) % n;
powersOf10modn = (powersOf10modn*10) % n;
}
return anwser;
}
If you have a number like : xyz % n , you only need to do yz % n.
You need n%10 + 1 last digits to make your operation with best performances.
Related
Given any number between 0 and 1, such as 0.84729347293923, is there a simple way to make it into 84729347293923 without string or regex manipulation? I can think of using a loop, which probably is no worse than using a string because it is O(n) with n being the number of digits. But is there a better way?
function getRandom() {
let r = Math.random();
while (Math.floor(r) !== r) r *= 10;
return r;
}
for (let i = 0; i < 10; i++)
console.log(getRandom());
Integers mod 1 = 0, non integers mod 1 != 0.
while ((r*=10) % 1);
Ok, just want to refactor my code (i realized that was bad so this is what i discovered to correctly get the value as you requested).
NOTE: As the question says that "given any number between 0 and 1", this solution only works for values between 0 and 1:
window.onload = ()=>{
function getLen(num){
let currentNumb = num;
let integratedArray = [];
let realLen = 0;
/*While the number is not an integer, we will multiply the copy of the original
*value by ten, and when the loop detects that the number is already an integer
*the while simply breaks, in this process we are storing each transformations
*of the number in an array called integratedArray*/
while(!(Number.isInteger(currentNumb))){
currentNumb *= 10;
integratedArray.push(currentNumb);
}
/*We iterate over the array and compare each value of the array with an operation
*in which the resultant value should be exactly the same as the actual item of the
*array, in the case that both are equal we assign the var realLen to i, and
*in case that the values were not the same, we simply breaks the loop, if the
*values are not the same, this indicates that we found the "trash numbers", so
*we simply skip them.*/
for(let i = 0; i < integratedArray.length; i++){
if(Math.floor(integratedArray[i]) === Math.floor(num * Math.pow(10, i + 1))){
realLen = i;
}else{
break;
}
}
return realLen;
}
//Get the float value of a number between 0 and 1 as an integer.
function getShiftedNumber(num){
//First we need the length to get the float part of the number as an integer
const len = getLen(num);
/*Once we have the length of the number we simply multiply the number by
*(10) ^ numberLength, this eliminates the comma (,), or point (.), and
*automatically transforms the number to an integer in this case a large integer*/
return num * (Math.pow(10, len));
}
console.log(getShiftedNumber(0.84729347293923));
}
So the explanation is the next:
Because we want to convert this number without using any string, regex or any another thing, first we need to get the length of the number, this is a bit hard to do without using string conversions... so i did the function getLen for this purpose.
In the function getLen, we have 3 variables:
currentNumb: This var is a copy of the original value (the original number), this value help us to found the length of the number and we can do some transforms to this value whitout changing the original reference of the number.
We need to multiply this value any times is needed to transform the number to an integer and then multiplyng this value by ten to ten.
with the help of a while (this method makes the number a false integer).
NOTE: I saw "False integer" because when i was making the tests i realized that in the number is being adding more digits than normal... (Very very strange), so this stupid but important thing makes neccesary the filter of these "trash numbers", so later we proccess them.
integratedArray: This array stores the values of the result of the first while operations, so the last number stored in this array is an integer, but this number is one of the "fake integers", so with this array we need to iterate later to compare what of those stored values are different to the original value multiplied by (10 * i + 1), so here is the hint:
In this case the first 12 values of this array are exactly the same with the operation of Math.floor(num * Math.pow(10, i + 1))), but in the 13th value of the array these values are not the same so... yes!, there are those "trash numbers" that we were searching for.
realLen: This is the variable where we will store the real length of the number converting the float part of this number in an integer.
Some binary search approach:
Its useless if avarage length < 8;
It contains floating point issues.
But hey it is O(log n) with tons of wasted side computations - i guess if one counts them its event worse than just plain multiplication.
I prefer #chiliNUT answer. One line stamp.
function floatToIntBinarySearch(number){
const max_safe_int_length = 16;
const powers = [
1,
10,
100,
1000,
10000,
100000,
1000000,
10000000,
100000000,
1000000000,
10000000000,
100000000000,
1000000000000,
10000000000000,
100000000000000,
1000000000000000,
10000000000000000
]
let currentLength = 16
let step = 16
let _number = number * powers[currentLength]
while(_number % 1 != 0 || (_number % 10 | 0) == 0){
step /= 2
if( (_number % 10 | 0) == 0 && !(_number % 1 != 0)){
currentLength = currentLength - step;
} else {
currentLength = step + currentLength;
}
if(currentLength < 1 || currentLength > max_safe_int_length * 2) throw Error("length is weird: " + currentLength)
_number = number * powers[currentLength]
console.log(currentLength, _number)
if(Number.isNaN(_number)) throw Error("isNaN: " + ((number + "").length - 2) + " maybe greater than 16?")
}
return number * powers[currentLength]
}
let randomPower = 10 ** (Math.random() * 10 | 0)
let test = (Math.random() * randomPower | 0) / randomPower
console.log(test)
console.log(floatToIntBinarySearch(test))
well this is problem number 12 on projecteuler website:
The sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first ten terms would be:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
Let us list the factors of the first seven triangle numbers:
1: 1
3: 1,3
6: 1,2,3,6
10: 1,2,5,10
15: 1,3,5,15
21: 1,3,7,21
28: 1,2,4,7,14,28
We can see that 28 is the first triangle number to have over five divisors.
What is the value of the first triangle number to have over five hundred divisors?
and here's my code (I'm new to Javascript)
let num = 1;
let add= 1;
let divisors = [];
while (divisors.length < 500){
divisors = []
for(let i = 1; i <= num; i++){
if(num % i == 0){
divisors.push(i);
}
}
add ++;
num += add;
}
console.log(num - add)
this code run fine when I change the while loop condition to 300 or less.
and this code running on Intel i7 Q740 1.75GHz.
when I try it nothing shows up on console and my question is that's because of my CPU and lack of power or my code has issue? I wait about 20 mins and still nothing as result.
This code is not very efficient as #Vasil Dininski pointed out but you won't reach the max Integer you just have to wait a while for your program to calculate.
I would recommend to optimize your code e.g. by writing a simple function which returns your number of divisors for your current number.
This could look similar to this:
function numberOfDivisors(num) {
var numOfDivisors = 0;
var sqrtOfNum = Math.sqrt(num);
for(var i = 1; i <= sqrtOfNum; i++) {
if(num % i == 0) {
numOfDivisors += 2;
}
}
// if your number is a perfect square you have to reduce it by one
if(sqrtOfNum * sqrtOfNum == num) {
numOfDivisors--;
}
return numOfDivisors;
}
Then you could use this method in your while loop like this:
var maxNumOfDivisors = 500;
var num = 0;
var i = 1;
while(numberOfDivisors(num) < maxNumOfDivisors) {
num += i;
i++;
}
which would return you the correct triangular number.
Please also note that triangular numbers start at 0 which is why my num is 0.
As you might have noticed, this algorithm might be a bit brute-force. A better one would be combine a few things. Let's assume the number we're looking for is "n":
Find all prime numbers in the range [1, square root of n]. You will
be iterating over n, so the sieve of
eratosthenes
will help in terms of efficiency (you can memoize primes you've already found)
Given that any number can be expressed as a prime number to some power, multiplied by a prime number to some power, etc. you can find all the combinations of those primes to a power, which are divisors of n.
This would be a more efficient, albeit a more complicated way to find them. You can take a look at this quora answer for more details: https://www.quora.com/What-is-an-efficient-algorithm-to-find-divisors-of-any-number
If my calculation is not wrong, you add one more to each next divisor from previous divisor and you get final result.
let prev = 0;
for(let i=0; i < 500; i++) {
prev = prev + i;
}
console.log("AA ", prev);
A friend of mine takes a sequence of numbers from 1 to n (where n > 0)
Within that sequence, he chooses two numbers, a and b
He says that the product of a and b should be equal to the sum of all numbers in the sequence, excluding a and b
Given a number n, could you tell me the numbers he excluded from the sequence?
Have found the solution to this Kata from Code Wars but it times out (After 12 seconds) in the editor when I run it; any ideas as too how I should further optimize the nested for loop and or remove it?
function removeNb(n) {
var nArray = [];
var sum = 0;
var answersArray = [];
for (let i = 1; i <= n; i++) {
nArray.push(n - (n - i));
sum += i;
}
var length = nArray.length;
for (let i = Math.round(n / 2); i < length; i++) {
for (let y = Math.round(n / 2); y < length; y++) {
if (i != y) {
if (i * y === sum - i - y) {
answersArray.push([i, y]);
break;
}
}
}
}
return answersArray;
}
console.log(removeNb(102));
.as-console-wrapper { max-height: 100% !important; top: 0; }
I think there is no reason for calculating the sum after you fill the array, you can do that while filling it.
function removeNb(n) {
let nArray = [];
let sum = 0;
for(let i = 1; i <= n; i++) {
nArray.push(i);
sum += i;
}
}
And since there could be only two numbers a and b as the inputs for the formula a * b = sum - a - b, there could be only one possible value for each of them. So, there's no need to continue the loop when you find them.
if(i*y === sum - i - y) {
answersArray.push([i,y]);
break;
}
I recommend looking at the problem in another way.
You are trying to find two numbers a and b using this formula a * b = sum - a - b.
Why not reduce the formula like this:
a * b + a = sum - b
a ( b + 1 ) = sum - b
a = (sum - b) / ( b + 1 )
Then you only need one for loop that produces the value of b, check if (sum - b) is divisible by ( b + 1 ) and if the division produces a number that is less than n.
for(let i = 1; i <= n; i++) {
let eq1 = sum - i;
let eq2 = i + 1;
if (eq1 % eq2 === 0) {
let a = eq1 / eq2;
if (a < n && a != i) {
return [[a, b], [b, a]];
}
}
}
You can solve this in linear time with two pointers method (page 77 in the book).
In order to gain intuition towards a solution, let's start thinking about this part of your code:
for(let i = Math.round(n/2); i < length; i++) {
for(let y = Math.round(n/2); y < length; y++) {
...
You already figured out this is the part of your code that is slow. You are trying every combination of i and y, but what if you didn't have to try every single combination?
Let's take a small example to illustrate why you don't have to try every combination.
Suppose n == 10 so we have 1 2 3 4 5 6 7 8 9 10 where sum = 55.
Suppose the first combination we tried was 1*10.
Does it make sense to try 1*9 next? Of course not, since we know that 1*10 < 55-10-1 we know we have to increase our product, not decrease it.
So let's try 2*10. Well, 20 < 55-10-2 so we still have to increase.
3*10==30 < 55-3-10==42
4*10==40 < 55-4-10==41
But then 5*10==50 > 55-5-10==40. Now we know we have to decrease our product. We could either decrease 5 or we could decrease 10, but we already know that there is no solution if we decrease 5 (since we tried that in the previous step). So the only choice is to decrease 10.
5*9==45 > 55-5-9==41. Same thing again: we have to decrease 9.
5*8==40 < 55-5-8==42. And now we have to increase again...
You can think about the above example as having 2 pointers which are initialized to the beginning and end of the sequence. At every step we either
move the left pointer towards right
or move the right pointer towards left
In the beginning the difference between pointers is n-1. At every step the difference between pointers decreases by one. We can stop when the pointers cross each other (and say that no solution can be obtained if one was not found so far). So clearly we can not do more than n computations before arriving at a solution. This is what it means to say that the solution is linear with respect to n; no matter how large n grows, we never do more than n computations. Contrast this to your original solution, where we actually end up doing n^2 computations as n grows large.
Hassan is correct, here is a full solution:
function removeNb (n) {
var a = 1;
var d = 1;
// Calculate the sum of the numbers 1-n without anything removed
var S = 0.5 * n * (2*a + (d *(n-1)));
// For each possible value of b, calculate a if it exists.
var results = [];
for (let numB = a; numB <= n; numB++) {
let eq1 = S - numB;
let eq2 = numB + 1;
if (eq1 % eq2 === 0) {
let numA = eq1 / eq2;
if (numA < n && numA != numB) {
results.push([numA, numB]);
results.push([numB, numA]);
}
}
}
return results;
}
In case it's of interest, CY Aries pointed this out:
ab + a + b = n(n + 1)/2
add 1 to both sides
ab + a + b + 1 = (n^2 + n + 2) / 2
(a + 1)(b + 1) = (n^2 + n + 2) / 2
so we're looking for factors of (n^2 + n + 2) / 2 and have some indication about the least size of the factor. This doesn't necessarily imply a great improvement in complexity for the actual search but still it's kind of cool.
This is part comment, part answer.
In engineering terms, the original function posted is using "brute force" to solve the problem, iterating every (or more than needed) possible combinations. The number of iterations is n is large - if you did all possible it would be
n * (n-1) = bazillio n
Less is More
So lets look at things that can be optimized, first some minor things, I'm a little confused about the first for loop and nArray:
// OP's code
for(let i = 1; i <= n; i++) {
nArray.push(n - (n - i));
sum += i;
}
??? You don't really use nArray for anything? Length is just n .. am I so sleep deprived I'm missing something? And while you can sum a consecutive sequence of integers 1-n by using a for loop, there is a direct and easy way that avoids a loop:
sum = ( n + 1 ) * n * 0.5 ;
THE LOOPS
// OP's loops, not optimized
for(let i = Math.round(n/2); i < length; i++) {
for(let y = Math.round(n/2); y < length; y++) {
if(i != y) {
if(i*y === sum - i - y) {
Optimization Considerations:
I see you're on the right track in a way, cutting the starting i, y values in half since the factors . But you're iterating both of them in the same direction : UP. And also, the lower numbers look like they can go a little below half of n (perhaps not because the sequence start at 1, I haven't confirmed that, but it seems the case).
Plus we want to avoid division every time we start an instantiation of the loop (i.e set the variable once, and also we're going to change it). And finally, with the IF statements, i and y will never be equal to each other the way we're going to create the loops, so that's a conditional that can vanish.
But the more important thing is the direction of transversing the loops. The smaller factor low is probably going to be close to the lowest loop value (about half of n) and the larger factor hi is probably going to be near the value of n. If we has some solid math theory that said something like "hi will never be less than 0.75n" then we could make a couple mods to take advantage of that knowledge.
The way the loops are show below, they break and iterate before the hi and low loops meet.
Moreover, it doesn't matter which loop picks the lower or higher number, so we can use this to shorten the inner loop as number pairs are tested, making the loop smaller each time. We don't want to waste time checking the same pair of numbers more than once! The lower factor's loop will start a little below half of n and go up, and the higher factor's loop will start at n and go down.
// Code Fragment, more optimized:
let nHi = n;
let low = Math.trunc( n * 0.49 );
let sum = ( n + 1 ) * n * 0.5 ;
// While Loop for the outside (incrementing) loop
while( low < nHi ) {
// FOR loop for the inside decrementing loop
for(let hi = nHi; hi > low; hi--) {
// If we're higher than the sum, we exit, decrement.
if( hi * low + hi + low > sum ) {
continue;
}
// If we're equal, then we're DONE and we write to array.
else if( hi * low + hi + low === sum) {
answersArray.push([hi, low]);
low = nHi; // Note this is if we want to end once finding one pair
break; // If you want to find ALL pairs for large numbers then replace these low = nHi; with low++;
}
// And if not, we increment the low counter and restart the hi loop from the top.
else {
low++;
break;
}
} // close for
} // close while
Tutorial:
So we set the few variables. Note that low is set slightly less than half of n, as larger numbers look like they could be a few points less. Also, we don't round, we truncate, which is essentially "always rounding down", and is slightly better for performance, (though it dosenit matter in this instance with just the single assignment).
The while loop starts at the lowest value and increments, potentially all the way up to n-1. The hi FOR loop starts at n (copied to nHi), and then decrements until the factor are found OR it intercepts at low + 1.
The conditionals:
First IF: If we're higher than the sum, we exit, decrement, and continue at a lower value for the hi factor.
ELSE IF: If we are EQUAL, then we're done, and break for lunch. We set low = nHi so that when we break out of the FOR loop, we will also exit the WHILE loop.
ELSE: If we get here it's because we're less than the sum, so we need to increment the while loop and reset the hi FOR loop to start again from n (nHi).
I'm trying to create my own decimal to binary converter with the method of decrementing the inputted variable (decimal value), by dividing it by 2 and storing the remainder (like 2nd grade math remainder), which is always either 0 or 1. Each of the remainder values i thin should be stored in an array and I think maybe put in backwards so that the most significant digit is first in the array (this is because when decrementing the remainer values are filled in left to right). Soooo yea i dont really know how to store the remainder values in an array using a function
Thanks in advance and if something is confusing then feel free to ask because im not even sure if this is the best method of doing this its just what i came up with
function decimalToBinary(num) {
var bin = 0;
while (num > 0) {
bin = num % 2 + bin;
num >>= 1; // basically /= 2 without remainder if any
}
alert("That decimal in binary is " + bin);
}
Your code is almost correct. The main problem is that bin starts out as 0; when you add a digit, they are added numerically, so your code ends up just counting the binary 1s: in this manner, 10 is initial 0, and +1+0+1+0, resulting in 2. You want to handle it as a string: ""+1+0+1+0 results in 1010. So, the only needed change is:
var bin = "";
If you want to solve it using arrays, with minimal changes to your code, it would be:
function decimalToBinary(num) {
var bin = [];
while (num > 0) {
bin.unshift(num % 2);
num >>= 1; // basically /= 2 without remainder if any
}
alert("That decimal in binary is " + bin.join(''));
}
Here, I use .unshift to add an element to the head of the array (and renumbering the remaining elements); .join() to collect them all into a string.
Or this:
function decimalToBinary(num) {
var bin = [];
while (num > 0) {
bin[bin.length] = num % 2;
num >>= 1; // basically /= 2 without remainder if any
}
alert("That decimal in binary is " + bin.reverse().join(''));
}
This is not as good, but illustrates some more things you can do with arrays: taking their length, setting an arbitrary element, and flipping them around.
I have written a custom Decimal to Binary method:
function toBinary (input) {
let options = [1];
let max = 0;
let i = 1;
while(i) {
max = Math.pow(2, i);
if (max > input) break;
options.push(max);
i++;
}
let j = options.length;
let result = new Array(j);
result.fill("0");
while(j >= 0) {
if (options[j] <= input) {
result[j] = "1"
input = input - options[j];
}
j--;
}
return [...result].reverse().join("");
}
//Test the toBin method with built-in toString(2)
toBinary(100) === (100).toString(2) // true
toBinary(1) === (1).toString(2) // true
toBinary(128) === (128).toString(2) // true
The following code results by combing common values into one:
//3,6,9,12,15------
//5,10,15----------
//result: sum of (3,5,6,9,10,12,15) = 60
int = 0;
for(var i=1;i<16;i++){
if(i%3==0 || i%5==0){
int +=i;
}
}
alert(int);//60
But I wanted to get the output without combining:
//3,6,9,12,15------|should ressult:
//5,10,15----------|75
//result without combining: sum of (3,6,9,12,15,5,10,15) == 75
If you say I should do && instead of || then it will just result common sum i.e. 15 here in example.
So, how can I do?
This will do the trick:
int = 0;
for(var i=1;i<16;i++){
if(i%3==0){
int +=i;
}
if(i%5==0){
int +=i;
}
}
alert(int);//75
Or if you prefer a more compact solution:
int = 0;
for(var i=1;i<16;i++){
int += (i % 3 ? 0 : i) + (i % 5 ? 0 : i);
}
alert(int);//75
Basically, for values that are multiples of 15, you count them twice.
Hints regarding a solution without using a loop: We can break the problem down to just list summing all the multiples of n between a and b-1. This can be computed directly without going through each number by taking a look at the sequence it produces. For n=3, a=1, b=28, this is:
3, 6, 9, 12, 15, 18, 21, 24, 27
Notice now that 27+3=30, 24+6=30, 21+9=30, 18+12=30, and there's only one outlier in the middle, 15. So really all you need to know in order to solve this is the minimum element in the sequence, the maximum element, the number of elements, and the middle element if the the number of elements is odd. (In fact you don't even really need to know the middle number once you realize that it's precisely half of the sum the min and the max).
To help you out a bit more:
var max = Math.floor((b - 1) / n) * n;
var min = a + n - (a % n);
var count = (max - min) / n + 1;