Introduction

This kind of question is usually shorthand for finding the smallest X such that X % P = N. Once the modulo meaning is made explicit, the problem becomes simple number theory: every solution belongs to one residue class, and the smallest non-negative representative is immediate.

The Direct Mathematical Answer

If the condition is X % P = N, then every solution has the form:

X = kP + N

for some integer k, as long as N is a valid remainder for modulus P.

That gives the first rule:

• if 0 <= N < P, the minimum non-negative X is N
• if N is outside that range, there is no integer X whose remainder is exactly N under the usual modulo definition

Examples:

• for P = 5 and N = 2, the smallest X is 2
• for P = 12 and N = 5, the smallest X is 5
• for P = 7 and N = 9, there is no valid solution because 9 is not a legal remainder modulo 7

Why the Answer Is So Short

Modulo arithmetic groups integers into repeating classes. All numbers congruent to N modulo P differ by multiples of P.

So for P = 4 and N = 1, the full solution set is:

• '1'
• '5'
• '9'
• '13'
• and so on

The smallest non-negative member of that class is 1, so that is the answer.

A small Python helper makes the rule explicit:

1def minimum_x(modulus, remainder):
2    if modulus <= 0:
3        raise ValueError("modulus must be positive")
4    if 0 <= remainder < modulus:
5        return remainder
6    raise ValueError("remainder is not valid for the given modulus")
7
8print(minimum_x(5, 2))
9print(minimum_x(12, 5))

This prints 2 and 5.

Watch for Problem Statement Ambiguity

Some exercises quietly assume X must be positive rather than non-negative. That changes one edge case.

If N = 0:

• minimum non-negative X is 0
• minimum positive X is P

You can encode that choice directly:

1def minimum_positive_x(modulus, remainder):
2    if modulus <= 0:
3        raise ValueError("modulus must be positive")
4    if not (0 <= remainder < modulus):
5        raise ValueError("remainder is not valid for the given modulus")
6    if remainder == 0:
7        return modulus
8    return remainder
9
10print(minimum_positive_x(7, 0))
11print(minimum_positive_x(7, 3))

This prints 7 and 3.

That distinction matters in programming contests and interview questions because authors do not always say whether zero counts.

Connection to Programming

In code, this pattern shows up whenever work wraps around a cycle, such as:

• indexing circular buffers
• scheduling repeating tasks
• hashing into buckets
• stepping through periodic states

If you know the desired remainder class, the smallest representative is the first valid state in that cycle.

For example, finding the first non-negative number whose remainder mod 8 is 6 is exactly the same as asking for the first position in an eight-step cycle that lands on state 6.

Common Pitfalls

The most common mistake is forgetting that a remainder must be in the standard range 0 through P - 1 when P is positive. Asking for X % 7 = 9 is invalid under that definition.

Another mistake is ignoring whether X may be zero. If the problem asks for the smallest positive solution, N = 0 no longer maps to 0.

Developers also sometimes mix mathematical modulo with language-specific behavior for negative operands. Many languages define % on negative numbers in implementation-specific or language-specific ways, so be careful if the problem allows negative values.

Finally, do not overcomplicate this particular task. Once the expression is interpreted as a modulo equation, the minimum non-negative solution is usually just the remainder itself.

Summary

• Interpreting the question as X % P = N gives a simple residue-class problem.
• If 0 <= N < P, the smallest non-negative solution is X = N.
• If N is not a valid remainder, there is no solution under the usual modulo definition.
• If the problem requires positive X, the N = 0 case becomes X = P.
• Most confusion comes from ambiguous wording, not from difficult math.