Find K-Beauty

You are given a number $num and a digit $k.

Write a script to find the K-Beauty of $num. The K-Beauty of an integer number is defined as the number of substrings of $num when it is read as a string which have a length of $k and are integer divisors of $num.

Examples

Example 1
Input: $num = 240, $k = 2
Output: 2
Substring with length 2:
24: 240 is divisible by 24
40: 240 is divisible by 40

Example 2
Input: $num = 1020, $k = 2
Output: 3
Substring with length 2:
10: 240 is divisible by 10
02: 240 is divisible by 2
20: 240 is divisible by 20

Example 3
Input: $num = 444, $k = 2
Output: 0
Substring with length 2:
First '44': 444 is not divisible by 44
Second '44': 444 is not divisible by 44

Example 4
Input: $num = 17, $k = 2
Output: 1
Substring with length 2:
17: 17 is divisible by 17

Example 5
Input: $num = 123, $k = 1
Output: 2
Substring with length 1:
1: 123 is divisible by 1
2: 123 is not divisible by 2
3: 123 is divisible by 3

Analysis

This is not hard to do with a simple search over all possible substrings of the stated length, which completes in negligible time for any reasonable length number. The only slight wrinkle is that substrings which are all zeroes - eg 000 - must be skipped.

Most K-beautiful numbers contain zeroes, and that tendency increases as $k increases and as the K-beauty, ie the number of divisors, increases.

Try it 

#!/usr/bin/perl

# Blog: http://ccgi.campbellsmiths.force9.co.uk/challenge

use v5.26;    # The Weekly Challenge - 2026-05-25
use utf8;     # Week 375 - task 2 - Find K-Beauty
use warnings; # Peter Campbell Smith
binmode STDOUT, ':utf8';
use Encode;

find_KBeauty(240, 2);
find_KBeauty(1001, 2);
find_KBeauty(100100, 4);
find_KBeauty(2001600, 5);
find_KBeauty(10250, 2);
find_KBeauty(160320, 2);
find_KBeauty(1015050, 3);

sub find_KBeauty {
    
    my ($number, $k, $l, $sub, @subs, $s);
    
    # initialise
    ($number, $k) = @_;
    $l = length($number);
    
    # loop over substrings and test for beauty
    for $s (0 .. $l - $k) {
        $sub = substr($number, $s, $k);
        next unless $sub > 0;
        push @subs, $sub if $number / $sub == int($number / $sub);
    }
    
    # report    
    say qq[\nInput:  \$number = $number, \$k = $k];
    say qq[Output: ] . scalar @subs .
        (scalar @subs ? q[ - ] . join(q[, ], @subs) : '');
}

11 lines of code

Output from script

Input:  $number = 240, $k = 2
Output: 2 - 24, 40

Input:  $number = 1001, $k = 2
Output: 1 - 01

Input:  $number = 100100, $k = 4
Output: 3 - 1001, 0010, 0100

Input:  $number = 2001600, $k = 5
Output: 3 - 20016, 00160, 01600

Input:  $number = 10250, $k = 2
Output: 4 - 10, 02, 25, 50

Input:  $number = 160320, $k = 2
Output: 5 - 16, 60, 03, 32, 20

Input:  $number = 1015050, $k = 3
Output: 5 - 101, 015, 150, 505, 050