3622. Check Divisibility by Digit Sum and Product

Difficulty: Easy

Topics: Mid Level, Math, Weekly Contest 459

You are given a positive integer n. Determine whether n is divisible by the sum of the following two values:

The digit sum of n (the sum of its digits).
The digit product of n (the product of its digits).

Return true if n is divisible by this sum; otherwise, return false.

Example 1:

Input: n = 99
Output: true
Explanation: Since 99 is divisible by the sum (9 + 9 = 18) plus product (9 * 9 = 81) of its digits (total 99), the output is true.

Example 2:

Input: n = 23
Output: false
Explanation: Since 23 is not divisible by the sum (2 + 3 = 5) plus product (2 * 3 = 6) of its digits (total 11), the output is false.

Example 3:

Input: n = 1
Output: false

Example 4:

Input: n = 2
Output: false

Example 5:

Input: n = 11
Output: false

Example 6:

Input: n = 12
Output: false

Example 7:

Input: n = 22
Output: false

Example 8:

Input: n = 24
Output: false

Example 9:

Input: n = 27
Output: false

Example 10:

Input: n = 33
Output: false

Example 11:

Input: n = 36
Output: false

Example 12:

Input: n = 48
Output: false

Example 13:

Input: n = 101
Output: false

Example 14:

Input: n = 111
Output: false

Example 15:

Input: n = 123
Output: false

Example 16:

Input: n = 222
Output: false

Example 17:

Input: n = 224
Output: false

Example 18:

Input: n = 315
Output: false

Example 19:

Input: n = 1000
Output: true

Example 20:

Input: n = 1000000
Output: true

Example 21:

Input: n = 999999
Output: false

Constraints:

1 <= n <= 10⁶

Hint:

Compute the digits' sum and product, then check if n % (sum + product) == 0.

Solution:

We implement a straightforward digit manipulation solution that computes both the sum and product of all digits in the given number n, then checks if n is divisible by their combined value. Our approach handles the constraints efficiently with a single pass through the digits using modulo and division operations.

Approach

Digit Extraction: Use a while loop to extract each digit from right to left using modulo operation (% 10)
Accumulate Values: Maintain running totals for digit sum (starting at 0) and digit product (starting at 1)
Handle Single Digits: Initialize product as 1 to handle single-digit numbers correctly (product = the digit itself)
Divisibility Check: Calculate divisor = sum + product and check if n % divisor == 0
Edge Cases: Since constraints specify n >= 1, we don't need to handle zero digits specially

Let's implement this solution in PHP: 3622. Check Divisibility by Digit Sum and Product

<?php
/**
 * @param Integer $n
 * @return Boolean
 */
function checkDivisibility(int $n): bool
{
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
echo checkDivisibility(99) .  "\n";         // Output: true
echo checkDivisibility(23) .  "\n";         // Output: false
echo checkDivisibility(1) .  "\n";          // Output: false
echo checkDivisibility(5) .  "\n";          // Output: false
echo checkDivisibility(11) .  "\n";         // Output: false
echo checkDivisibility(12) .  "\n";         // Output: false
echo checkDivisibility(22) .  "\n";         // Output: false
echo checkDivisibility(24) .  "\n";         // Output: false
echo checkDivisibility(27) .  "\n";         // Output: false
echo checkDivisibility(33) .  "\n";         // Output: false
echo checkDivisibility(36) .  "\n";         // Output: false
echo checkDivisibility(48) .  "\n";         // Output: false
echo checkDivisibility(101) .  "\n";        // Output: false
echo checkDivisibility(111) .  "\n";        // Output: false
echo checkDivisibility(123) .  "\n";        // Output: false
echo checkDivisibility(222) .  "\n";        // Output: false
echo checkDivisibility(224) .  "\n";        // Output: false
echo checkDivisibility(315) .  "\n";        // Output: false
echo checkDivisibility(1000) .  "\n";       // Output: true
echo checkDivisibility(1000000) .  "\n";    // Output: true
echo checkDivisibility(999999) .  "\n";     // Output: false
?>

Explanation:

Digit Sum Calculation: We repeatedly take the last digit using num % 10 and add it to $sum, then remove the last digit by integer division intval($num / 10)
Digit Product Calculation: We multiply each extracted digit into $product. Starting $product at 1 ensures correct multiplication (multiplying by 1 doesn't change the result)
Single-Digit Handling: For numbers like 5, the loop extracts digit 5, sets $sum = 5, $product = 5, so divisor = 10, and 5 % 10 != 0 (except for 1 where 1 % 2 != 0)
Zero Digits: Not possible in this problem since n >= 1 and we only extract actual digits (no leading zeros). However, if a digit were 0, the product would become 0, making the divisor equal to the sum
Divisibility Test: The final check n % (sum + product) == 0 determines if n is evenly divisible by the combined value
Return Value: Returns true if divisible, false otherwise, matching the required boolean output

Complexity Analysis

Time Complexity: O(d) where d is the number of digits in n. Since n ≤ 10⁶, d ≤ 7, making this effectively O(1) constant time
Space Complexity: O(1) - we use only a few integer variables regardless of input size

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