Question

Suppose n = 10,000, a = 10,023, and b = 10,004. Use an identity of modular arithmetic to calculate in your head (a • b) mod n.

Short answer

(a • b) mod n = 92.

Step-by-step solution

Understand Modular Arithmetic Properties

One key property of modular arithmetic is that: \((a \cdot b) \mod n = [(a \mod n) \cdot (b \mod n)] \mod n\). This allows us to simplify calculations by reducing \(a\) and \(b\) under modulo \(n\) before multiplying.

Simplify 'a' Using Modulo

Calculate \(a \mod n\). Given \(a = 10,023\) and \(n = 10,000\), the remainder when \(10,023\) is divided by \(10,000\) is \(23\). Therefore, \(a \equiv 23 \mod 10,000\).

Simplify 'b' Using Modulo

Calculate \(b \mod n\). Given \(b = 10,004\) and \(n = 10,000\), the remainder when \(10,004\) is divided by \(10,000\) is \(4\). Therefore, \(b \equiv 4 \mod 10,000\).

Multiply Simplified Results

Now, use the reduced numbers: \((a \cdot b) \mod n = (23 \cdot 4) \mod n\). First, calculate \(23 \cdot 4 = 92\).

Apply Modulo Operation to the Result

With the result from the multiplication, calculate \(92 \mod 10,000\). Since 92 is already less than 10,000, \(92 \equiv 92 \mod 10,000\). Thus, the result is \(92\).

Key concepts

Understanding the Modulo Operation

The modulo operation is a fundamental concept in modular arithmetic. It finds the remainder when one integer is divided by another. When we say "\(a \mod n\)", we are asking what the remainder is when \(a\) is divided by \(n\). This helps simplify calculations, particularly in problems involving large numbers.

• The process essentially reduces numbers, which can simplify further mathematical operations.
• If \(a = nq + r\), where \(q\) is the quotient and \(r\) the remainder, then \(a \equiv r \mod n\).

For instance, given \(a = 10,023\) and \(n = 10,000\), the remainder is \(23\). So, \(10,023 \equiv 23 \mod 10,000\). Similarly, for \(b = 10,004\), the calculation yields \(b \equiv 4 \mod 10,000\).
Using this concept leads to a simple and effective means to handle large numbers in calculations.

Mathematical Properties of Modular Arithmetic

Modular arithmetic isn't just a tool for simplifying calculations; it also has several mathematical properties that can be very useful. One of these properties is the ability to multiply numbers and take the modulus efficiently using an identity:\[(a \cdot b) \mod n = [(a \mod n) \cdot (b \mod n)] \mod n\]This principle allows us to break down larger operations into smaller, more manageable parts. In the exercise, we used this property to simplify (10,023 \times 10,004) mod 10,000.

• The reduction of \(a\) and \(b\) by using \(mod n\) turns them into smaller numbers.
• It's easier to compute \((23 \times 4) \mod 10,000\) than \((10,023 \times 10,004) \mod 10,000\).

Thanks to this property, we get efficient and scalable calculations, even as \(a\) or \(b\) grow large.

Number Theory and Modular Arithmetic

Number theory, a branch of pure mathematics, often utilizes modular arithmetic. It deals with integers and properties such as divisibility, prime numbers, and congruences. Modular arithmetic is a critical tool in number theory because it helps manage large numbers elegantly.

• It has applications in cryptography, coding theory, and computer science.
• It simplifies many complex numerical problems into more digestible parts.

Through this lens, modular arithmetic helps explore deeper mathematical truths. This deep dive finds its roots in congruences, where we explore relationships like \(a \equiv b \mod n\). Such congruences become the stalwarts of number theory, making problems that seem challenging more approachable with modular arithmetic tools.