Question

Show that if \(a_{i} \equiv b_{i}(\bmod m)\) for \(i=1,2, \ldots, n\), where \(m\) is a positive integer and \(a_{i}, b_{i}\) are integers for \(j=1,2, \ldots, n\), then \(\sum_{i=1}^{n} a_{i} \equiv \sum_{i=1}^{n} b_{i}(\bmod m)\)

Short answer

Question: Prove that if \(a_i \equiv b_i \pmod{m}\) for all \(i = 1, 2, \ldots, n\), then \(\sum_{i=1}^n a_i \equiv \sum_{i=1}^n b_i \pmod{m}\). Solution: By using the definition of congruence modulo m and properties of modular arithmetic, we can show that \(\sum_{i=1}^{n} a_{i} \equiv \sum_{i=1}^{n} b_{i} \pmod{m}\) for all \(i = 1, 2, \ldots, n\).

Step-by-step solution

Understand the definition of congruence modulo m

Given a positive integer \(m\), two integers \(a\) and \(b\) are said to be congruent modulo \(m\) if their difference is a multiple of \(m\). This is expressed as \(a \equiv b \pmod{m}\). The definition implies that \(a - b = km\) for some integer \(k\). In our problem, we are given that \(a_i \equiv b_i \pmod{m}\) for all \(i=1, 2, \ldots, n\). This means that \(a_i - b_i = k_im\) for all \(i=1,2,\ldots,n\), where \(k_i\) is an integer.

Use properties of modular arithmetic

We will use the properties of modular arithmetic to simplify and manipulate the given congruences. One of the useful properties of modular arithmetic is that if \(a \equiv b \pmod{m}\) and \(c \equiv d \pmod{m}\), then \(a+c \equiv b+d \pmod{m}\). This property can also be extended to sums of more than two terms.

Apply the property to our problem

We want to show that \(\sum_{i=1}^{n} a_{i} \equiv \sum_{i=1}^{n} b_{i} \pmod{m}\). We will apply the property from Step 2 to the given congruences \(a_i \equiv b_i \pmod{m}\) for all \(i=1,2,\ldots,n\). As a result of applying the property, we can find that: \begin{align*} (a_1 + a_2 + \cdots + a_n) &\equiv (b_1+ b_2 + \cdots + b_n) \pmod{m} \\ \sum_{i=1}^{n} a_{i} &\equiv \sum_{i=1}^{n} b_{i} \pmod{m} \end{align*} Hence, we have shown that \(\sum_{i=1}^{n} a_{i} \equiv \sum_{i=1}^{n} b_{i} \pmod{m}\), as desired.

Key concepts

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, which considers the remainder when one number is divided by another, known as the modulus. Think of it as the mathematics of clocks. When you add hours on a clock, you go back to zero after reaching 12, starting a new cycle. This 'wrapping around' effect is at the core of modular arithmetic.

In formal terms, when we say that two numbers, such as a and b, are congruent modulo m, we are saying that when divided by m, they leave the same remainder. This can also be understood as both numbers being an equal distance apart, but in multiples of m. The notation we use to express this relationship is a ≡ b (mod m).

For instance, the statement 14 ≡ 3 (mod 11) means that when 14 and 3 are divided by 11, the remainder is the same. Thus, 14 and 3 are 'equivalent' in modulus 11. These calculations are beneficial because they allow us to work with large numbers by simplifying them into smaller, more manageable ones that carry the same properties relevant to the modulus.

Integer Congruence

Integer congruence, a fundamental concept in modular arithmetic, involves the relationship between two integers with respect to a given modulus. When we state that a is congruent to b modulo m, written as a ≡ b (mod m), it's another way of saying that a and b have no difference that matters within the modular system defined by m. In other words, they could be separated by a multiple of m.

For example, if we take 17 and 5, and we are working modulo 6, these two numbers would be considered congruent because both leave a remainder of 5 when divided by 6. The calculation 17 - 5 = 12, which is a multiple of 6, reaffirms their congruence. Integer congruence allows us to group numbers into classes where each member of the class represents a possible remainder when divided by the modulus.

Arithmetic Properties

Arithmetic properties in modular arithmetic are similar to those in regular arithmetic, but with a special focus on the remainders. Some key properties include:

• Closure: Addition, subtraction, and multiplication of integers modulo m always result in an integer modulo m.
• Commutativity: For all integers a and b, a + b ≡ b + a (mod m) and ab ≡ ba (mod m).
• Associativity: For all integers a, b, and c, (a + b) + c ≡ a + (b + c) (mod m) and (ab)c ≡ a(bc) (mod m).
• Distributive: The distributive property of multiplication over addition is preserved in modular arithmetic, i.e., a(b + c) ≡ ab + ac (mod m).

Understanding these arithmetic properties is crucial when operating within modular systems because they guarantee the operations stay consistent with regular arithmetic, maintaining their intuitive nature while confined to the modular structure. When we used these properties in our original exercise, we effortlessly expanded the principle of congruence from individual integers to an entire sum, demonstrating the coherence and power of modular arithmetic in simplifying complex operations.