Question 1:
If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000.
Program 1:
// This Program is quite EASY and does not need any application.
public class Problem_1{
public static void main(String []args){
int sum = 0;
for(int i = 3;i<1000;i++){
if((i%3 == 0) || (i%5 == 0)){
sum += i;
}
}
System.out.println(sum);
}
}
Question 2:
Program 2:
public class Problem_2 {
public static void main(String[] args) {
int sum = 0;
for(int i=1;;i++){
int fibonacciNumber = fibonacci(i);
if(fibonacciNumber > 4000000){
break;
}
if((fibonacciNumber % 2) == 0){
sum += fibonacciNumber;
}
}
System.out.println(sum);
}
public static int fibonacci(int num){
if(num == 1){
return(1);
}
else if(num == 2){
return(2);
}
else{
return(fibonacci(num-1)+fibonacci(num - 2));
}
}
}
Question 3 :
Program 3 :
public class Problem_3 {
public static void main(String[] args) {
long n = 600851475143L;
while(true){
long p = smallestPrimeFactor(n);
//We divide the number till the number becomes indivisible(prime number)
// As we begin from the smallest prime factor , last one is the largest one
if(p < n){
n /= p;
p = smallestPrimeFactor(n);
}
else{
break;
}
}
System.out.println(n);
}
//This method will always return the smallest prime factor of a number
private static long smallestPrimeFactor(long num){
for(long i=2,end = (long)Math.sqrt(num);i<=end;i++){
if(num % i == 0){
return(i);
}
}
return(num);
}
}
Question 4 :
Program 4:
// This program really does not need any explanation. Let me know if you are //stuck any where.
public class Problem_4 {
public static void main(String[] args) {
int maxP = -1;
for(int i=100;i<1000;i++){
for(int j = 100;j<1000;j++){
int p = i * j;
if(maxP < p && isPalindrome(p)){
maxP = p;
}
}
}
System.out.println(maxP);
}
private static boolean isPalindrome(int num){
String number = Integer.toString(num);
StringBuffer sb = new StringBuffer(number);
return(number.equals(sb.reverse().toString()));
}
}
Question 5 :
Program 5 :
public class Problem_5 {
public static void main(String[] args) {
//You need to understand the BigInteger Class of Java for this Program
//Please let me know if you need the explanation
BigInteger allcm = BigInteger.ONE;
for(int i = 1;i<=20;i++){
allcm = lcm(BigInteger.valueOf(i),allcm);
}
System.out.println(allcm.toString());
}
//Below method is the simple logic to find the LCM
private static BigInteger lcm(BigInteger x,BigInteger y){
return(x.divide(x.gcd(y)).multiply(y));
}
}
Question 6 :
public class Program_6 {
private static final int N = 100;
public static void main(String[] args) {
//For Mathematician
int sum = N * (N + 1) / 2;
int sum2 = N * (N + 1) * (2 * N + 1) / 6;
System.out.println((sum * sum ) - sum2);
int summ = 0,summ2 = 0;
//For just a programmer
for(int i = 0;i < 100;i++){
summ += i;
summ2 += (i * i);
}
System.out.println((sum * sum ) - sum2);
}
}
Question 7 :
Program 7 :
// It's a quite easy problem.
public class Program_7 {
If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000.
Program 1:
// This Program is quite EASY and does not need any application.
public class Problem_1{
public static void main(String []args){
int sum = 0;
for(int i = 3;i<1000;i++){
if((i%3 == 0) || (i%5 == 0)){
sum += i;
}
}
System.out.println(sum);
}
}
Question 2:
Each new term in the Fibonacci sequence is generated by adding the
previous two terms. By starting with 1 and 2, the first 10 terms will
be:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
By considering the terms in the Fibonacci sequence whose values do
not exceed four million, find the sum of the even-valued terms.Program 2:
public class Problem_2 {
public static void main(String[] args) {
int sum = 0;
for(int i=1;;i++){
int fibonacciNumber = fibonacci(i);
if(fibonacciNumber > 4000000){
break;
}
if((fibonacciNumber % 2) == 0){
sum += fibonacciNumber;
}
}
System.out.println(sum);
}
public static int fibonacci(int num){
if(num == 1){
return(1);
}
else if(num == 2){
return(2);
}
else{
return(fibonacci(num-1)+fibonacci(num - 2));
}
}
}
Question 3 :
The prime factors of 13195 are 5, 7, 13 and 29.
What is the largest prime factor of the number 600851475143 ?
What is the largest prime factor of the number 600851475143 ?
Program 3 :
public class Problem_3 {
public static void main(String[] args) {
long n = 600851475143L;
while(true){
long p = smallestPrimeFactor(n);
//We divide the number till the number becomes indivisible(prime number)
// As we begin from the smallest prime factor , last one is the largest one
if(p < n){
n /= p;
p = smallestPrimeFactor(n);
}
else{
break;
}
}
System.out.println(n);
}
//This method will always return the smallest prime factor of a number
private static long smallestPrimeFactor(long num){
for(long i=2,end = (long)Math.sqrt(num);i<=end;i++){
if(num % i == 0){
return(i);
}
}
return(num);
}
}
Question 4 :
A palindromic number reads the same both ways. The largest palindrome
made from the product of two 2-digit numbers is 9009 = 91 × 99.
Find the largest palindrome made from the product of two 3-digit numbers.
Find the largest palindrome made from the product of two 3-digit numbers.
Program 4:
// This program really does not need any explanation. Let me know if you are //stuck any where.
public class Problem_4 {
public static void main(String[] args) {
int maxP = -1;
for(int i=100;i<1000;i++){
for(int j = 100;j<1000;j++){
int p = i * j;
if(maxP < p && isPalindrome(p)){
maxP = p;
}
}
}
System.out.println(maxP);
}
private static boolean isPalindrome(int num){
String number = Integer.toString(num);
StringBuffer sb = new StringBuffer(number);
return(number.equals(sb.reverse().toString()));
}
}
Question 5 :
2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.
What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?
What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?
Program 5 :
public class Problem_5 {
public static void main(String[] args) {
//You need to understand the BigInteger Class of Java for this Program
//Please let me know if you need the explanation
BigInteger allcm = BigInteger.ONE;
for(int i = 1;i<=20;i++){
allcm = lcm(BigInteger.valueOf(i),allcm);
}
System.out.println(allcm.toString());
}
//Below method is the simple logic to find the LCM
private static BigInteger lcm(BigInteger x,BigInteger y){
return(x.divide(x.gcd(y)).multiply(y));
}
}
Question 6 :
The sum of the squares of the first ten natural numbers is,
Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.
12 + 22 + ... + 102 = 385
The square of the sum of the first ten natural numbers is,
(1 + 2 + ... + 10)2 = 552 = 3025
Hence the difference between the sum of the squares of the first ten
natural numbers and the square of the sum is 3025 − 385 = 2640.Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.
public class Program_6 {
private static final int N = 100;
public static void main(String[] args) {
//For Mathematician
int sum = N * (N + 1) / 2;
int sum2 = N * (N + 1) * (2 * N + 1) / 6;
System.out.println((sum * sum ) - sum2);
int summ = 0,summ2 = 0;
//For just a programmer
for(int i = 0;i < 100;i++){
summ += i;
summ2 += (i * i);
}
System.out.println((sum * sum ) - sum2);
}
}
Question 7 :
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.
What is the 10 001st prime number?
What is the 10 001st prime number?
Program 7 :
// It's a quite easy problem.
public class Program_7 {
public static void main(String[] args) {
int num = 2,count = 0;
while(true){
if(isPrime(num)){
count++;
if(count == 10001){
break;
}
}
num++;
}
System.out.println(num);
}
private static boolean isPrime(int num){
if(num == 1){
return(false);
}
if(num == 2){
return(true);
}
if(num % 2 == 0){
return(false);
}
for(int i=3,end = (int)Math.sqrt(num);i <= end;i++){
if(num % i == 0){
return(false);
}
}
return(true);
}
}
Question 8 :
The four adjacent digits in the 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832.
73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450
Find the thirteen adjacent digits in the 1000-digit number that have the greatest product. What is the value of this product?96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450
Program 8:
public class Problem_8 {
private static String NUMBER = "7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088055111254069874715852386305071569329096329522744304355766896648950445244523161731856403098711121722383113622298934233803081353362766142828064444866452387493035890729629049156044077239071381051585930796086670172427121883998797908792274921901699720888093776657273330010533678812202354218097512545405947522435258490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637048440319989000889524345065854122758866688116427171479924442928230863465674813919123162824586178664583591245665294765456828489128831426076900422421902267105562632111110937054421750694165896040807198403850962455444362981230987879927244284909188845801561660979191338754992005240636899125607176060588611646710940507754100225698315520005593572972571636269561882670428252483600823257530420752963450";
private static final int ADJACENT = 13;
public static void main(String[] args) {
long max = -1;
for(int i = 0; (i + ADJACENT) < NUMBER.length();i++){
long product = 1;
for(int j = 0;j < ADJACENT;j++){
product *= NUMBER.charAt(i+j) - '0';
}
max = Math.max(product,max);
}
System.out.println(max);
}
}
Question 9 :
A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,
There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.
a2 + b2 = c2
For example, 32 + 42 = 9 + 16 = 25 = 52.There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.
Program 9 :
public class Problem_9 {
public static void main(String[] args) {
int result = productOfTriplet();
if(result == -1){
System.out.println("NOT FOUND");
}
else{
System.out.println(result);
}
}
private static int productOfTriplet(){
for(int a = 1;a<1000;a++){
for(int b = a + 1;b < 1000;b++){
int c = 1000 - a - b;
if((a * a + b * b) == (c * c)){
return(a * b * c);
}
}
}
return(-1);
}
}
Question 10 :
The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
Find the sum of all the primes below two million.
Find the sum of all the primes below two million.
Program 10 :
public class Program_10 {
private static final int LIMIT = 2000000;
public static void main(String[] args) {
boolean []isPrime = listPrimality();
long sum = 0;
for(int i = 2;i<LIMIT;i++){
if(isPrime[i]){
sum += i;
}
}
System.out.println(sum);
}
private static boolean[] listPrimality(){
boolean []isPrime = new boolean[LIMIT + 1];
isPrime[2] = true;
for(int i = 3; i < LIMIT;i+=2){
isPrime[i] = true;
}
for(int i = 3,end = (int)Math.sqrt(LIMIT);i <= end;i+=2){
if(isPrime[i]){
for(int j = (i * i);j <= LIMIT;j += i << 1){
isPrime[j] = false;
}
}
}
return(isPrime);
}
}
Question 11 :
In the 20×20 grid below, four numbers along a diagonal line have been marked in red.
08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00
81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65
52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91
22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80
24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50
32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70
67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21
24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72
21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95
78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92
16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57
86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58
19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40
04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66
88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69
04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36
20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16
20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54
01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48
The product of these numbers is 26 × 63 × 78 × 14 = 1788696.
What is the greatest product of four adjacent numbers in the same direction (up, down, left, right, or diagonally) in the 20×20 grid?
Program 11 :
public class Problem_11 {
public static void main(String[] args) {
int max = -1;
max = Math.max(maxProduct(1,0),max);
max = Math.max(maxProduct(0,1),max);
max = Math.max(maxProduct(1,1),max);
max = Math.max(maxProduct(1,-1),max);
System.out.println(max);
}
private static int maxProduct(int dx,int dy){
int max = -1;
for(int x = 0;x < GRIDOFNUMBERS.length;x++){
for(int y = 0;y < GRIDOFNUMBERS[x].length;y++){
max = Math.max(getProduct(x, y, dx, dy, 4),max);
}
}
return(max);
}
//This method returns the product if the boundary condition is satisfied
private static int getProduct(int x,int y,int dx,int dy,int n){
int product = 1;
if(!(boundaryChecks(x + (n - 1) * dx, y + (n - 1) * dy))){
return (-1);
}
for(int i = 0;i < n;x+=dx,y+=dy,i++){
product *= GRIDOFNUMBERS[x][y];
}
return(product);
}
//This method is for checking the boundary.x is checking rows and y is checking columns
private static boolean boundaryChecks(int x,int y){
return(x >= 0 && x < GRIDOFNUMBERS.length && y >= 0 && y < GRIDOFNUMBERS[x].length);
}
private static int [][]GRIDOFNUMBERS = {
{ 8, 2,22,97,38,15, 0,40, 0,75, 4, 5, 7,78,52,12,50,77,91, 8},
{49,49,99,40,17,81,18,57,60,87,17,40,98,43,69,48, 4,56,62, 0},
{81,49,31,73,55,79,14,29,93,71,40,67,53,88,30, 3,49,13,36,65},
{52,70,95,23, 4,60,11,42,69,24,68,56, 1,32,56,71,37, 2,36,91},
{22,31,16,71,51,67,63,89,41,92,36,54,22,40,40,28,66,33,13,80},
{24,47,32,60,99, 3,45, 2,44,75,33,53,78,36,84,20,35,17,12,50},
{32,98,81,28,64,23,67,10,26,38,40,67,59,54,70,66,18,38,64,70},
{67,26,20,68, 2,62,12,20,95,63,94,39,63, 8,40,91,66,49,94,21},
{24,55,58, 5,66,73,99,26,97,17,78,78,96,83,14,88,34,89,63,72},
{21,36,23, 9,75, 0,76,44,20,45,35,14, 0,61,33,97,34,31,33,95},
{78,17,53,28,22,75,31,67,15,94, 3,80, 4,62,16,14, 9,53,56,92},
{16,39, 5,42,96,35,31,47,55,58,88,24, 0,17,54,24,36,29,85,57},
{86,56, 0,48,35,71,89, 7, 5,44,44,37,44,60,21,58,51,54,17,58},
{19,80,81,68, 5,94,47,69,28,73,92,13,86,52,17,77, 4,89,55,40},
{ 4,52, 8,83,97,35,99,16, 7,97,57,32,16,26,26,79,33,27,98,66},
{88,36,68,87,57,62,20,72, 3,46,33,67,46,55,12,32,63,93,53,69},
{ 4,42,16,73,38,25,39,11,24,94,72,18, 8,46,29,32,40,62,76,36},
{20,69,36,41,72,30,23,88,34,62,99,69,82,67,59,85,74, 4,36,16},
{20,73,35,29,78,31,90, 1,74,31,49,71,48,86,81,16,23,57, 5,54},
{ 1,70,54,71,83,51,54,69,16,92,33,48,61,43,52, 1,89,19,67,48},
};
}
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