Introduction

In this article, we explore the solution to a problem involving a number that satisfies specific remainder conditions when divided by certain integers. The problem requires understanding of congruences and modular arithmetic. We will walk through the solution step-by-step, providing explanations and mathematical derivations to ensure clarity and comprehension.

Solving the Problem

The goal is to find the least number x that meets the following criteria:

x equiv 2 pmod{356810} and x equiv 2 pmod{12} x equiv 0 pmod{13}

This problem can be approached using the Least Common Multiple (LCM) of the divisors and modular arithmetic techniques.

Step 1: Find the Least Common Multiple (LCM)

The first step involves finding the LCM of the divisors 356810, 12, and 356810 and 12.

Prime factorize the numbers: 356810 2 x 5 x 71362 12 2^2 x 3

Take the highest powers of each prime factor to find the LCM:

2^2 x 3 x 5 x 71362

Simplifying, we find:

LCM 4 x 3 x 5 x 71362 4281720

Step 2: Set up the congruence

Since x equiv 2 pmod{356810} and x equiv 2 pmod{12}, we can express x as:

x 4281720k 2

Step 3: Solve for the second condition

We need to incorporate the second condition x equiv 0 pmod{13}:

4281720k 2 equiv 0 pmod{13}

First, reduce 4281720 pmod{13}:

4281720 div 13 329363 remainder 1

So, 4281720 equiv 1 pmod{13}

The congruence becomes:

1k 2 equiv 0 pmod{13}

1k equiv -2 pmod{13}

-2 is equivalent to 11 pmod{13}:

1k equiv 11 pmod{13}

Step 4: Find the multiplicative inverse of 1 modulo 13

To solve for k, we need to find the multiplicative inverse of 1 modulo 13, which is 1 itself. So, k 11.

Step 5: Substitute back to find x

Substitute k 11 back into the equation for x:

x 4281720(11) 2 47108922 2 47108924

Step 6: Find the least positive solution

The least number that satisfies both conditions is:

boxed{47108924}

Conclusion

In this article, we have demonstrated how to find the least number that meets specific remainder conditions using congruences and modular arithmetic. The solution involves finding the LCM, setting up congruences, and solving for the multiplicative inverse.