Solving Congruences: The Least Number with Specific Remainders

Modular arithmetic, commonly known as congruences, finds extensive applications in various fields, including cryptography, number theory, and computer science. This article will explore the least positive integer that satisfies a set of congruences, specifically those that leave certain remainders when divided by specific numbers.

Introduction to Congruences

A congruence is an equation stating that two numbers are congruent modulo n if their difference is a multiple of n. For example, x ≡ a (mod n) means that when x is divided by n, the remainder is a.

Problem Statement

Find the least positive integer x that satisfies the following congruences:

x ≡ 2 (mod 5) x ≡ 3 (mod 6) x ≡ 4 (mod 7)

Step-by-Step Solution

Step 1: Solve the First Two Congruences

First, we solve for x in the first two congruences:

x ≡ 2 (mod 5) x ≡ 3 (mod 6)

From x ≡ 2 (mod 5), we can write:

x 5k 2 for some integer k

Substituting this into the second congruence, we get:

5k 2 ≡ 3 (mod 6) Simplifying, we have:

5k ≡ 1 (mod 6)

Since 5 ≡ -1 (mod 6), we can rewrite the congruence as:

-k ≡ 1 (mod 6) or k ≡ -1 (mod 6) which simplifies to:

k ≡ 5 (mod 6)

Thus, we can express k as:

k 6j 5 for some integer j

Substituting back into our expression for x, we get:

x 5(6j 5) 2 30j 25 2 30j 27

This simplifies to:

x ≡ 27 (mod 30)

Step 2: Solve with the Third Congruence

Now, we incorporate the third congruence:

x ≡ 4 (mod 7)

Substituting x 30j 27 into the third congruence, we get:

30j 27 ≡ 4 (mod 7)

Simplifying, we note that:

30 ≡ 2 (mod 7) and 27 ≡ 6 (mod 7)

Thus, the congruence becomes:

2j 6 ≡ 4 (mod 7)

Subtracting 6 from both sides, we get:

2j ≡ -2 (mod 7) or 2j ≡ 5 (mod 7)

Multiplying both sides by the multiplicative inverse of 2 modulo 7 (which is 4), since 2*4 ≡ 1 (mod 7), we get:

j ≡ 20 (mod 7) or j ≡ 6 (mod 7)

We can express j as:

j 7m 6 for some integer m

Substituting back into our expression for x, we get:

x 30(7m 6) 27 210m 180 27 210m 207

This simplifies to:

x ≡ 207 (mod 210)

Step 3: Find the Minimum Positive Solution

The minimum positive solution is obtained by setting m 0, giving:

x 207

Therefore, the least positive integer that satisfies the given congruences is:

boxed{207}

Further Exploration

It's interesting to note that there are many such integers that satisfy the same congruences, of the form 207 210p, where p is any integer. These form a sequence such as 207, 417, 627, and so on.

This exploration of congruences and their solutions, particularly when leaving specific remainders, showcases the elegance and practical applications of modular arithmetic.