LeetCode #2310 — MEDIUM

Sum of Numbers With Units Digit K

Move from brute-force thinking to an efficient approach using math strategy.

Solve on LeetCode
The Problem

Problem Statement

Given two integers num and k, consider a set of positive integers with the following properties:

  • The units digit of each integer is k.
  • The sum of the integers is num.

Return the minimum possible size of such a set, or -1 if no such set exists.

Note:

  • The set can contain multiple instances of the same integer, and the sum of an empty set is considered 0.
  • The units digit of a number is the rightmost digit of the number.

Example 1:

Input: num = 58, k = 9
Output: 2
Explanation:
One valid set is [9,49], as the sum is 58 and each integer has a units digit of 9.
Another valid set is [19,39].
It can be shown that 2 is the minimum possible size of a valid set.

Example 2:

Input: num = 37, k = 2
Output: -1
Explanation: It is not possible to obtain a sum of 37 using only integers that have a units digit of 2.

Example 3:

Input: num = 0, k = 7
Output: 0
Explanation: The sum of an empty set is considered 0.

Constraints:

  • 0 <= num <= 3000
  • 0 <= k <= 9
Patterns Used

Roadmap

  1. Brute Force Baseline
  2. Core Insight
  3. Algorithm Walkthrough
  4. Edge Cases
  5. Full Annotated Code
  6. Interactive Study Demo
  7. Complexity Analysis
Step 01

Brute Force Baseline

Problem summary: Given two integers num and k, consider a set of positive integers with the following properties: The units digit of each integer is k. The sum of the integers is num. Return the minimum possible size of such a set, or -1 if no such set exists. Note: The set can contain multiple instances of the same integer, and the sum of an empty set is considered 0. The units digit of a number is the rightmost digit of the number.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Math · Dynamic Programming · Greedy

Example 1

58
9

Example 2

37
2

Example 3

0
7

Related Problems

  • Digit Count in Range (digit-count-in-range)
  • Count Integers With Even Digit Sum (count-integers-with-even-digit-sum)
  • Sum of Number and Its Reverse (sum-of-number-and-its-reverse)
Step 02

Core Insight

What unlocks the optimal approach

  • Try solving this recursively.
  • Create a method that takes an integer x as a parameter. This method returns the minimum possible size of a set where each number has units digit k and the sum of the numbers in the set is x.
Interview move: turn each hint into an invariant you can check after every iteration/recursion step.
Step 03

Algorithm Walkthrough

Iteration Checklist

  1. Define state (indices, window, stack, map, DP cell, or recursion frame).
  2. Apply one transition step and update the invariant.
  3. Record answer candidate when condition is met.
  4. Continue until all input is consumed.
Use the first example testcase as your mental trace to verify each transition.
Step 04

Edge Cases

Minimum Input
Single element / shortest valid input
Validate boundary behavior before entering the main loop or recursion.
Duplicates & Repeats
Repeated values / repeated states
Decide whether duplicates should be merged, skipped, or counted explicitly.
Extreme Constraints
Upper-end input sizes
Re-check complexity target against constraints to avoid time-limit issues.
Invalid / Corner Shape
Empty collections, zeros, or disconnected structures
Handle special-case structure before the core algorithm path.
Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #2310: Sum of Numbers With Units Digit K
class Solution {
    public int minimumNumbers(int num, int k) {
        if (num == 0) {
            return 0;
        }
        for (int i = 1; i <= num; ++i) {
            int t = num - k * i;
            if (t >= 0 && t % 10 == 0) {
                return i;
            }
        }
        return -1;
    }
}
Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Press Step or Run All to begin.
Step 07

Complexity Analysis

Time
O(n × m)
Space
O(n × m)

Approach Breakdown

RECURSIVE
O(2ⁿ) time
O(n) space

Pure recursion explores every possible choice at each step. With two choices per state (take or skip), the decision tree has 2ⁿ leaves. The recursion stack uses O(n) space. Many subproblems are recomputed exponentially many times.

DYNAMIC PROGRAMMING
O(n × m) time
O(n × m) space

Each cell in the DP table is computed exactly once from previously solved subproblems. The table dimensions determine both time and space. Look for the state variables — each unique combination of state values is one cell. Often a rolling array can reduce space by one dimension.

Shortcut: Count your DP state dimensions → that’s your time. Can you drop one? That’s your space optimization.
Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.

State misses one required dimension

Wrong move: An incomplete state merges distinct subproblems and caches incorrect answers.

Usually fails on: Correctness breaks on cases that differ only in hidden state.

Fix: Define state so each unique subproblem maps to one DP cell.

Using greedy without proof

Wrong move: Locally optimal choices may fail globally.

Usually fails on: Counterexamples appear on crafted input orderings.

Fix: Verify with exchange argument or monotonic objective before committing.