# Ugly Numbers

Ugly numbers are numbers whose only prime factors are 2, 3 or 5. The sequence 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, … shows the first 11 ugly numbers. By convention, 1 is included.

Given a number n, the task is to find n’th Ugly number.

Examples:

```Input  : n = 7
Output : 8

Input  : n = 10
Output : 12

Input  : n = 15
Output : 24

Input  : n = 150
Output : 5832
```

## Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

Method 1 (Simple)
Loop for all positive integers until ugly number count is smaller than n, if an integer is ugly than increment ugly number count.

To check if a number is ugly, divide the number by greatest divisible powers of 2, 3 and 5, if the number becomes 1 then it is an ugly number otherwise not.

For example, let us see how to check for 300 is ugly or not. Greatest divisible power of 2 is 4, after dividing 300 by 4 we get 75. Greatest divisible power of 3 is 3, after dividing 75 by 3 we get 25. Greatest divisible power of 5 is 25, after dividing 25 by 25 we get 1. Since we get 1 finally, 300 is ugly number.

Implementation:

## C/C++

 `// CPP program to find nth ugly number ` `# include ` `# include ` ` `  `/*This function divides a by greatest divisible  ` `  ``power of b*/` `int` `maxDivide(``int` `a, ``int` `b) ` `{ ` `  ``while` `(a%b == 0) ` `   ``a = a/b;  ` `  ``return` `a; ` `}     ` ` `  `/* Function to check if a number is ugly or not */` `int` `isUgly(``int` `no) ` `{ ` `  ``no = maxDivide(no, 2); ` `  ``no = maxDivide(no, 3); ` `  ``no = maxDivide(no, 5); ` `   `  `  ``return` `(no == 1)? 1 : 0; ` `}     ` ` `  `/* Function to get the nth ugly number*/` `int` `getNthUglyNo(``int` `n) ` `{ ` `  ``int` `i = 1;  ` `  ``int` `count = 1;   ``/* ugly number count */`  ` `  `  ``/*Check for all integers untill ugly count  ` `    ``becomes n*/`  `  ``while` `(n > count) ` `  ``{ ` `    ``i++;       ` `    ``if` `(isUgly(i)) ` `      ``count++;  ` `  ``} ` `  ``return` `i; ` `} ` ` `  `/* Driver program to test above functions */` `int` `main() ` `{ ` `    ``unsigned no = getNthUglyNo(150); ` `    ``printf``(``"150th ugly no. is %d "``,  no); ` `    ``getchar``(); ` `    ``return` `0; ` `} `

## Java

 `// Java program to find nth ugly number ` `class` `GFG { ` `     `  `    ``/*This function divides a by greatest ` `    ``divisible power of b*/` `    ``static` `int` `maxDivide(``int` `a, ``int` `b) ` `    ``{ ` `        ``while``(a % b == ``0``) ` `            ``a = a/b; ` `        ``return` `a; ` `    ``} ` `     `  `    ``/* Function to check if a number  ` `    ``is ugly or not */` `    ``static` `int` `isUgly(``int` `no) ` `    ``{ ` `        ``no = maxDivide(no, ``2``); ` `        ``no = maxDivide(no, ``3``); ` `        ``no = maxDivide(no, ``5``); ` `         `  `        ``return` `(no == ``1``)? ``1` `: ``0``; ` `    ``} ` `     `  `    ``/* Function to get the nth ugly  ` `    ``number*/` `    ``static` `int` `getNthUglyNo(``int` `n) ` `    ``{ ` `        ``int` `i = ``1``; ` `         `  `        ``// ugly number count  ` `        ``int` `count = ``1``;  ` `         `  `        ``// check for all integers  ` `        ``// until count becomes n  ` `        ``while``(n > count) ` `        ``{ ` `            ``i++; ` `            ``if``(isUgly(i) == ``1``) ` `                ``count++; ` `        ``} ` `        ``return` `i; ` `    ``} ` `     `  `    ``/* Driver program to test above ` `    ``functions */` `    ``public` `static` `void` `main(String args[]) ` `    ``{ ` `        ``int` `no = getNthUglyNo(``150``); ` `        ``System.out.println(``"150th ugly "` `                       ``+ ``"no. is "``+ no); ` `    ``} ` `} ` ` `  `// This code has been contributed by  ` `// Amit Khandelwal (Amit Khandelwal 1) `

## Python3

 `# Python3 code to find nth ugly number ` ` `  `# This function divides a by greatest  ` `# divisible power of b ` `def` `maxDivide( a, b ): ` `    ``while` `a ``%` `b ``=``=` `0``: ` `        ``a ``=` `a ``/` `b ` `    ``return` `a  ` ` `  `# Function to check if a number  ` `# is ugly or not ` `def` `isUgly( no ): ` `    ``no ``=` `maxDivide(no, ``2``) ` `    ``no ``=` `maxDivide(no, ``3``) ` `    ``no ``=` `maxDivide(no, ``5``) ` `    ``return` `1` `if` `no ``=``=` `1` `else` `0` ` `  `# Function to get the nth ugly number ` `def` `getNthUglyNo( n ): ` `    ``i ``=` `1` `    ``count ``=` `1` `# ugly number count ` ` `  `    ``# Check for all integers untill  ` `    ``# ugly count becomes n ` `    ``while` `n > count: ` `        ``i ``+``=` `1` `        ``if` `isUgly(i): ` `            ``count ``+``=` `1` `    ``return` `i ` ` `  `# Driver code to test above functions ` `no ``=` `getNthUglyNo(``150``) ` `print``(``"150th ugly no. is "``, no) ` ` `  `# This code is contributed by "Sharad_Bhardwaj". `

## C#

 `// C# program to find nth ugly number ` `using` `System; ` ` `  `class` `GFG { ` ` `  `    ``/*This function divides a by  ` `    ``greatest divisible power of b*/` `    ``static` `int` `maxDivide(``int` `a, ``int` `b) ` `    ``{ ` `        ``while``(a % b == 0) ` `            ``a = a / b; ` `        ``return` `a; ` `    ``} ` `     `  `    ``/* Function to check if a number  ` `    ``is ugly or not */` `    ``static` `int` `isUgly(``int` `no) ` `    ``{ ` `        ``no = maxDivide(no, 2); ` `        ``no = maxDivide(no, 3); ` `        ``no = maxDivide(no, 5); ` `         `  `        ``return` `(no == 1)? 1 : 0; ` `    ``} ` `     `  `    ``/* Function to get the nth ugly ` `    ``number*/` `    ``static` `int` `getNthUglyNo(``int` `n) ` `    ``{ ` `        ``int` `i = 1; ` `         `  `        ``// ugly number count  ` `        ``int` `count = 1;  ` `         `  `        ``// check for all integers ` `        ``// until count becomes n  ` `        ``while``(n > count) ` `        ``{ ` `            ``i++; ` `            ``if``(isUgly(i) == 1) ` `                ``count++; ` `        ``} ` `        ``return` `i; ` `    ``} ` `     `  `    ``// Driver code ` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int` `no = getNthUglyNo(150); ` `         `  `        ``Console.WriteLine(``"150th ugly"` `                  ``+ ``" no. is "` `+ no); ` `    ``} ` `} ` ` `  `// This code is contributed by Sam007. `

## PHP

 ` ``\$count``) ` `{ ` `    ``\$i``++;      ` `    ``if` `(isUgly(``\$i``)) ` `    ``\$count``++;  ` `} ` `return` `\$i``; ` `} ` ` `  `    ``// Driver Code ` `    ``\$no` `= getNthUglyNo(150); ` `    ``echo` `"150th ugly no. is "``. ``\$no``; ` ` `  `// This code is contributed by Sam007 ` `?> `

Output:

`150th ugly no. is 5832 `

This method is not time efficient as it checks for all integers until ugly number count becomes n, but space complexity of this method is O(1)

Method 2 (Use Dynamic Programming)
Here is a time efficient solution with O(n) extra space. The ugly-number sequence is 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, …
because every number can only be divided by 2, 3, 5, one way to look at the sequence is to split the sequence to three groups as below:
(1) 1×2, 2×2, 3×2, 4×2, 5×2, …
(2) 1×3, 2×3, 3×3, 4×3, 5×3, …
(3) 1×5, 2×5, 3×5, 4×5, 5×5, …

We can find that every subsequence is the ugly-sequence itself (1, 2, 3, 4, 5, …) multiply 2, 3, 5. Then we use similar merge method as merge sort, to get every ugly number from the three subsequence. Every step we choose the smallest one, and move one step after.

```1 Declare an array for ugly numbers:  ugly[n]
2 Initialize first ugly no:  ugly = 1
3 Initialize three array index variables i2, i3, i5 to point to
1st element of the ugly array:
i2 = i3 = i5 =0;
4 Initialize 3 choices for the next ugly no:
next_mulitple_of_2 = ugly[i2]*2;
next_mulitple_of_3 = ugly[i3]*3
next_mulitple_of_5 = ugly[i5]*5;
5 Now go in a loop to fill all ugly numbers till 150:
For (i = 1; i < 150; i++ )
{
/* These small steps are not optimized for good
readability. Will optimize them in C program */
next_ugly_no  = Min(next_mulitple_of_2,
next_mulitple_of_3,
next_mulitple_of_5);

ugly[i] =  next_ugly_no

if (next_ugly_no  == next_mulitple_of_2)
{
i2 = i2 + 1;
next_mulitple_of_2 = ugly[i2]*2;
}
if (next_ugly_no  == next_mulitple_of_3)
{
i3 = i3 + 1;
next_mulitple_of_3 = ugly[i3]*3;
}
if (next_ugly_no  == next_mulitple_of_5)
{
i5 = i5 + 1;
next_mulitple_of_5 = ugly[i5]*5;
}

}/* end of for loop */
6.return next_ugly_no
```

Example:
Let us see how it works

```initialize
ugly[] =  | 1 |
i2 =  i3 = i5 = 0;

First iteration
ugly = Min(ugly[i2]*2, ugly[i3]*3, ugly[i5]*5)
= Min(2, 3, 5)
= 2
ugly[] =  | 1 | 2 |
i2 = 1,  i3 = i5 = 0  (i2 got incremented )

Second iteration
ugly = Min(ugly[i2]*2, ugly[i3]*3, ugly[i5]*5)
= Min(4, 3, 5)
= 3
ugly[] =  | 1 | 2 | 3 |
i2 = 1,  i3 =  1, i5 = 0  (i3 got incremented )

Third iteration
ugly = Min(ugly[i2]*2, ugly[i3]*3, ugly[i5]*5)
= Min(4, 6, 5)
= 4
ugly[] =  | 1 | 2 | 3 |  4 |
i2 = 2,  i3 =  1, i5 = 0  (i2 got incremented )

Fourth iteration
ugly = Min(ugly[i2]*2, ugly[i3]*3, ugly[i5]*5)
= Min(6, 6, 5)
= 5
ugly[] =  | 1 | 2 | 3 |  4 | 5 |
i2 = 2,  i3 =  1, i5 = 1  (i5 got incremented )

Fifth iteration
ugly = Min(ugly[i2]*2, ugly[i3]*3, ugly[i5]*5)
= Min(6, 6, 10)
= 6
ugly[] =  | 1 | 2 | 3 |  4 | 5 | 6 |
i2 = 3,  i3 =  2, i5 = 1  (i2 and i3 got incremented )

Will continue same way till I < 150
```

## C/C++

 `// C++ program to find n'th Ugly number ` `# include ` `using` `namespace` `std; ` ` `  `/* Function to get the nth ugly number*/` `unsigned getNthUglyNo(unsigned n) ` `{ ` `    ``unsigned ugly[n]; ``// To store ugly numbers ` `    ``unsigned i2 = 0, i3 = 0, i5 = 0; ` `    ``unsigned next_multiple_of_2 = 2; ` `    ``unsigned next_multiple_of_3 = 3; ` `    ``unsigned next_multiple_of_5 = 5; ` `    ``unsigned next_ugly_no = 1; ` ` `  `    ``ugly = 1; ` `    ``for` `(``int` `i=1; i

## Java

 `// Java program to find nth ugly number ` `import` `java.lang.Math; ` ` `  `class` `UglyNumber ` `{ ` `    ``/* Function to get the nth ugly number*/` `    ``int` `getNthUglyNo(``int` `n) ` `    ``{ ` `        ``int` `ugly[] = ``new` `int``[n];  ``// To store ugly numbers ` `        ``int` `i2 = ``0``, i3 = ``0``, i5 = ``0``; ` `        ``int` `next_multiple_of_2 = ``2``; ` `        ``int` `next_multiple_of_3 = ``3``; ` `        ``int` `next_multiple_of_5 = ``5``; ` `        ``int` `next_ugly_no = ``1``; ` `         `  `        ``ugly[``0``] = ``1``; ` `         `  `        ``for``(``int` `i = ``1``; i < n; i++) ` `        ``{ ` `            ``next_ugly_no = Math.min(next_multiple_of_2, ` `                                  ``Math.min(next_multiple_of_3, ` `                                        ``next_multiple_of_5)); ` `             `  `            ``ugly[i] = next_ugly_no; ` `            ``if` `(next_ugly_no == next_multiple_of_2) ` `            ``{ ` `               ``i2 = i2+``1``; ` `               ``next_multiple_of_2 = ugly[i2]*``2``; ` `            ``} ` `            ``if` `(next_ugly_no == next_multiple_of_3) ` `            ``{ ` `               ``i3 = i3+``1``; ` `               ``next_multiple_of_3 = ugly[i3]*``3``; ` `            ``} ` `            ``if` `(next_ugly_no == next_multiple_of_5) ` `            ``{ ` `               ``i5 = i5+``1``; ` `               ``next_multiple_of_5 = ugly[i5]*``5``; ` `            ``} ` `        ``} ``/*End of for loop (i=1; i

## Python

 `# Python program to find n'th Ugly number ` ` `  `# Function to get the nth ugly number ` `def` `getNthUglyNo(n): ` ` `  `    ``ugly ``=` `[``0``] ``*` `n ``# To store ugly numbers ` ` `  `    ``# 1 is the first ugly number ` `    ``ugly[``0``] ``=` `1` ` `  `    ``# i2, i3, i5 will indicate indices for 2,3,5 respectively ` `    ``i2 ``=` `i3 ``=``i5 ``=` `0` ` `  `    ``# set initial multiple value ` `    ``next_multiple_of_2 ``=` `2` `    ``next_multiple_of_3 ``=` `3` `    ``next_multiple_of_5 ``=` `5` ` `  `    ``# start loop to find value from ugly to ugly[n] ` `    ``for` `l ``in` `range``(``1``, n): ` ` `  `        ``# choose the min value of all available multiples ` `        ``ugly[l] ``=` `min``(next_multiple_of_2, next_multiple_of_3, next_multiple_of_5) ` ` `  `        ``# increment the value of index accordingly ` `        ``if` `ugly[l] ``=``=` `next_multiple_of_2: ` `            ``i2 ``+``=` `1` `            ``next_multiple_of_2 ``=` `ugly[i2] ``*` `2` ` `  `        ``if` `ugly[l] ``=``=` `next_multiple_of_3: ` `            ``i3 ``+``=` `1` `            ``next_multiple_of_3 ``=` `ugly[i3] ``*` `3` ` `  `        ``if` `ugly[l] ``=``=` `next_multiple_of_5:  ` `            ``i5 ``+``=` `1` `            ``next_multiple_of_5 ``=` `ugly[i5] ``*` `5` ` `  `    ``# return ugly[n] value ` `    ``return` `ugly[``-``1``] ` ` `  `def` `main(): ` ` `  `    ``n ``=` `150` ` `  `    ``print` `getNthUglyNo(n) ` ` `  ` `  `if` `__name__ ``=``=` `'__main__'``: ` `    ``main() ` ` `  `#This code is contributed by Neelam Yadav `

## C#

 `// C# program to count inversions in an array ` `using` `System; ` `using` `System.Collections.Generic; ` ` `  `class` `GFG { ` ` `  `    ``/* Function to get the nth ugly number*/` `    ``static` `int` `getNthUglyNo(``int` `n) ` `    ``{ ` `         `  `        ``// To store ugly numbers ` `        ``int` `[]ugly = ``new` `int``[n];  ` `        ``int` `i2 = 0, i3 = 0, i5 = 0; ` `        ``int` `next_multiple_of_2 = 2; ` `        ``int` `next_multiple_of_3 = 3; ` `        ``int` `next_multiple_of_5 = 5; ` `        ``int` `next_ugly_no = 1; ` `         `  `        ``ugly = 1; ` `         `  `        ``for``(``int` `i = 1; i < n; i++) ` `        ``{ ` `            ``next_ugly_no = Math.Min(next_multiple_of_2, ` `                           ``Math.Min(next_multiple_of_3, ` `                                  ``next_multiple_of_5)); ` `             `  `            ``ugly[i] = next_ugly_no; ` `             `  `            ``if` `(next_ugly_no == next_multiple_of_2) ` `            ``{ ` `                ``i2 = i2 + 1; ` `                ``next_multiple_of_2 = ugly[i2] * 2; ` `            ``} ` `             `  `            ``if` `(next_ugly_no == next_multiple_of_3) ` `            ``{ ` `                ``i3 = i3 + 1; ` `                ``next_multiple_of_3 = ugly[i3] * 3; ` `            ``} ` `            ``if` `(next_ugly_no == next_multiple_of_5) ` `            ``{ ` `                ``i5 = i5 + 1; ` `                ``next_multiple_of_5 = ugly[i5] * 5; ` `            ``} ` `        ``}  ` `         `  `        ``return` `next_ugly_no; ` `    ``} ` `     `  `    ``// Driver code ` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int` `n = 150; ` `        ``Console.WriteLine(getNthUglyNo(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by Sam007 `

## PHP

 ` `

Output :

```5832
```

Time Complexity: O(n)
Auxiliary Space: O(n)

Super Ugly Number (Number whose prime factors are in given set)

Please write comments if you find any bug in the above program or other ways to solve the same problem.

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