Question

Are \(9 x^{3}+2 x\) and \(x^{2}-4\) congruent modulo \(2 ?\)

Short answer

No, the expressions \(9 x^{3}+2 x\) and \(x^{2}-4\) are not congruent modulo 2.

Step-by-step solution

Understand the Concept of Modulo and Congruency

Two numbers are said to be congruent modulo n if they both have the same remainder when divided by n. In this exercise we are required to determine if the expressions \(9 x^{3}+2 x\) and \(x^{2}-4\) are congruent modulo 2, this means that if these expressions are divided by 2, they will have to give the same remainders.

Break Down the Expressions

Consider the first expression, \(9 x^{3}+2 x\), its constituents are \(9 x^{3}\) and \(2 x\). Similarly, for the second expression \(x^{2}-4\), the constituents are \(x^{2}\) and \(-4\). When we divide all these constituents by 2, the remainder should be same for each of them if these expressions are congruent modulo 2.

Apply the Modulo Operation

In \(9x^{3}\), if we consider 9 modulo 2, we obtain 1 as the remainder. Thus \(9x^{3}\) modulo 2 results in \(1*x^{3}\). For \(2x\), 2 modulo 2 is 0 thus \(2x\) modulo 2 is \(0\). Next, consider the second expression \(x^{2}-4\). For \(x^{2}\), as we don't have a constant to reduce, it remains \(x^{2}\) modulo 2 after applying modulo operation. For \(-4\), \(-4\) modulo 2 results in 0. Therefore both expressions modulo 2 will be \(x^{3}\) and \(x^{2}\).

Check for Congruency

If both expressions were congruent modulo 2, they should have produced similar expressions as result after applying modulo operation. But, as seen in the result from applying modulo operation, \(9 x^{3}+2 x\) modulo 2 is \(x^{3}\) and \(x^{2}-4\) modulo 2 is \(x^{2}\). So, \(9 x^{3}+2 x\) and \(x^{2}-4\) are not congruent modulo 2 as their results are not the same.

Key concepts

Modulo Operation

The modulo operation is a fundamental concept in mathematics that deals with the remainder after division. When you perform the operation a modulo n, represented as a mod n, you are essentially dividing a by n and taking the remainder of this division as the result. This operation is crucial for understanding concepts like clock arithmetic, where the hours reset after reaching a certain number instead of continuing indefinitely.

For example, if we consider the number 7 modulo 3, the calculation would be 7 divided by 3, which equals 2 with a remainder of 1. So, 7 modulo 3 is 1. This operation is used not only in mathematics but also in computer science for hashing and cryptography.

Congruency in Mathematics

In mathematics, congruency is a relation that describes the equivalence of numbers with respect to the remainder when divided by a common divisor. Formally, we say that two numbers, a and b, are congruent modulo n, written as a ≡ b (mod n), if they have the same remainder when divided by n.

Understanding Congruency with an Example

Let's look at the numbers 8 and 14. Are they congruent modulo 3? Dividing both by 3, we get remainders of 2 and 2, respectively. Since the remainders are equal, 8 and 14 are congruent modulo 3: 8 ≡ 14 (mod 3).

This concept is not only theoretical but has practical applications in areas such as computer science, where it can simplify complex computations by considering numbers within a smaller, more manageable set of residues.

Divisibility and Remainders

Divisibility is a property that reflects how a number can be evenly divided by another without leaving a remainder. When we talk about remainders, we are referring to what is left over after this division. Divisibility and remainders go hand in hand in determining the outcome of the modulo operation and in characterizing the relationship between numbers under the concept of congruence.

For instance, if we divide 15 by 4, the quotient is 3, and the remainder is 3 (15 = 4 * 3 + 3). Here, 15 is not divisible by 4 since it does not divide evenly without a remainder. The remainder is a crucial element in the study of number theory, encryption algorithms, and coding theory in computer science.

Importance of the Remainder

The remainder is central to understanding the modulo operation and congruence. It ensures that any integer can be compared to any other with respect to a modulus. This makes it invaluable in designing cyclical processes or in situations where the concept of 'wrap around' is important, such as in modular arithmetic or cyclic groups.