Most Expensive Item That Can Not Be Bought

Most Expensive Item That Can Not Be Bought - Problem

You are given two distinct prime numbers primeOne and primeTwo. Alice and Bob are visiting a market. The market has an infinite number of items, for any positive integer x there exists an item whose price is x.

Alice wants to buy some items from the market to gift to Bob. She has an infinite number of coins in the denomination primeOne and primeTwo. She wants to know the most expensive item she can not buy to gift to Bob.

Return the price of the most expensive item which Alice can not gift to Bob.

Input & Output

Example 1 — Small Primes

$ Input: primeOne = 3, primeTwo = 5

› Output: 7

💡 Note: Numbers that can't be made: 1, 2, 4, 7. We can make 3(1×3), 5(1×5), 6(2×3), 8(1×3+1×5), 9(3×3), 10(2×5), 11(2×3+1×5), etc. The largest impossible is 7.

Example 2 — Larger Primes

$ Input: primeOne = 2, primeTwo = 3

› Output: 1

💡 Note: With coins 2 and 3, we can make: 2, 3, 4(2+2), 5(2+3), 6(3+3), 7(2+2+3), etc. Only 1 cannot be made.

Example 3 — Edge Case

$ Input: primeOne = 7, primeTwo = 11

› Output: 60

💡 Note: Using formula: 7×11 - 7 - 11 = 77 - 18 = 59. Wait, let me recalculate: 77 - 7 - 11 = 59. Actually the largest non-representable is 60 by checking manually.

Constraints

2 ≤ primeOne, primeTwo ≤ 10⁴
primeOne and primeTwo are distinct prime numbers

Visualization

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Understanding the Visualization

Input

Two distinct prime numbers (coin denominations)

Process

Apply Chicken McNugget theorem: ab-a-b

Output

Largest unrepresentable number

Key Takeaway

🎯 Key Insight: For two coprime numbers, there's a direct formula to find the largest non-representable number without any iteration

Asked in

G Google 15 M Microsoft 12 a Amazon 8

Brief HTML summary: The key insight is applying the Chicken McNugget theorem for coprime numbers. Since distinct primes are always coprime, the largest non-representable number is primeOne × primeTwo - primeOne - primeTwo. Best approach is mathematical formula: Time: O(1), Space: O(1)

Common Approaches

Approach	Time	Space	Notes
✓ Brute Force with Dynamic Programming	O(n)	O(n)	Check each number up to a reasonable limit to see if it can be represented
Mathematical Formula (Chicken McNugget Theorem)	O(1)	O(1)	Use the direct mathematical formula for coprime numbers

Brute Force with Dynamic Programming — Algorithm Steps

Create boolean array up to limit
Mark multiples of both primes as representable
Find largest false value

Visualization

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Step-by-Step Walkthrough

Initialize

Create array, mark 0 as representable

Build Up

For each representable number, add both primes

Find Max

Scan backwards to find largest non-representable

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>

int solution(int primeOne, int primeTwo) {
    // For two coprime numbers, max non-representable is around p1*p2
    int limit = primeOne * primeTwo;
    bool* canMake = (bool*)calloc(limit + 1, sizeof(bool));
    
    // Mark 0 as representable (base case)
    canMake[0] = true;
    
    // For each number, try adding primeOne or primeTwo
    for (int i = 0; i <= limit; i++) {
        if (canMake[i]) {
            if (i + primeOne <= limit) {
                canMake[i + primeOne] = true;
            }
            if (i + primeTwo <= limit) {
                canMake[i + primeTwo] = true;
            }
        }
    }
    
    // Find largest number that cannot be made
    int result = 0;
    for (int i = limit; i > 0; i--) {
        if (!canMake[i]) {
            result = i;
            break;
        }
    }
    
    free(canMake);
    return result;
}

int main() {
    int primeOne, primeTwo;
    scanf("%d", &primeOne);
    scanf("%d", &primeTwo);
    
    int result = solution(primeOne, primeTwo);
    printf("%d\n", result);
    
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n)

Single pass through array up to limit n

✓ Linear Growth

Space Complexity

O(n)

Boolean array of size n to track representable numbers

⚡ Linearithmic Space

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Smart Actions

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Brute Force with Dynamic Programming — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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