Question

What is the least nonnegative integer \(f\) with \(f \equiv 2 \bmod 3, f \equiv 3 \bmod 5\), and \(f \equiv 2 \bmod 7\) ?

Short answer

The least nonnegative integer solution is 23.

Step-by-step solution

Identify the System of Congruences

The problem gives us a system of congruences that we need to solve:\[ f \equiv 2 \mod 3 \]\[ f \equiv 3 \mod 5 \]\[ f \equiv 2 \mod 7 \]

Solve the First and Third Congruences Together

Notice that the first and third congruences, \( f \equiv 2 \mod 3 \) and \( f \equiv 2 \mod 7 \), can be combined. The least positive solution to these combined congruences is the least common multiple of 3 and 7 subtracting 1,Thus we look for numbers in the form:\[ f = 21k + 2 \] (since \(21 = 3 \times 7\))

Substitute into the Second Congruence

Substitute the expression \( f = 21k + 2 \) into the second congruence:\[ 21k + 2 \equiv 3 \mod 5 \]Simplifying, we subtract 2 from both sides:\[ 21k \equiv 1 \mod 5 \]Since \(21 \equiv 1 \mod 5\), the congruence simplifies to \(k \equiv 1 \mod 5\).

Solve for k

The solution for \(k \equiv 1 \mod 5\) suggests \(k = 5m + 1\) for integer \(m\). Substitute back:\[ f = 21k + 2 = 21(5m + 1) + 2 = 105m + 23 \]

Calculate the Least Nonnegative Integer f

To find the least nonnegative integer solution, set \(m = 0\):\[ f = 105 \times 0 + 23 = 23 \]

Verify the Solution

Verify that 23 satisfies all the original congruences:\( 23 \mod 3 = 2 \), \( 23 \mod 5 = 3 \), and \( 23 \mod 7 = 2 \). This confirms that 23 is correct.

Key concepts

System of Congruences

In number theory, a system of congruences is a set of equations where each equation dictates that a certain remainder is obtained when a variable is divided by a respective divisor. These equations share a common variable. In our exercise, the system of congruences is as follows:

• \( f \equiv 2 \mod 3 \)
• \( f \equiv 3 \mod 5 \)
• \( f \equiv 2 \mod 7 \)

The challenge is to find an integer solution that satisfies all of these conditions simultaneously. These kinds of problems are fundamental in number theory and appear often in cryptography and other mathematical applications. Solving a system of congruences involves finding a number that fits all given conditions for each modulus, which can sometimes be accomplished through guesswork, substitution, or application of theorems.

Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) is a powerful tool in number theory used for solving systems of congruences with different moduli. When the moduli are pairwise coprime, as they are in our problem (i.e., 3, 5, and 7 are coprime), CRT ensures us there is a unique solution modulo the product of the moduli.To apply CRT effectively, we start by combining congruences with the same remainder. Our example involves combining:

• \( f \equiv 2 \mod 3 \)
• \( f \equiv 2 \mod 7 \)

Since the common remainder is 2, these two can be "merged," meaning the answer needs only satisfy the common least remainder over the product of these moduli, \(3 \times 7 = 21\). We represent this merged state as \( f = 21k + 2 \) and solve it concurrently with other congruences.Using CRT massively simplifies our system, allowing us to analyze otherwise complex remnants layer by layer.

Least Common Multiple

Least Common Multiple (LCM) is used to identify the smallest multiple that two or more numbers share. When combined with congruences, it helps simplify problems by reducing equivalent congruences into a single, more manageable one.In this exercise, the LCM of 3 and 7 is calculated because both have the same remainder 2 when the variable is considered. It aids in "merging" these congruences, thus mapping their common cycle.To find the LCM, we use the formula based on their greatest common divisor (GCD):\[\text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)}\]Here, since 3 and 7 are coprime, their LCM is simply \(3 \times 7 = 21\).With the LCM computed, our equation simplifies, revealing the merged behavior of the system: \( f \equiv 2 \mod 21 \). This relationship allows us to explore just one remainder across fewer conditions, streamlining our approach to find the solution.