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In mathematics, a congruence relation is a way to express that two numbers give the same remainder when divided by a given number, called the modulus.
For example, if we say that integers \(x\) and \(y\) are congruent modulo \(m\), we write it as \(x \equiv y \pmod{m}\). This means that when \(x\) and \(y\) are divided by \(m\), they both leave the same remainder.
For example, if \(x = 14\), \(y = 2\), and \(m = 6\), then \(14 \equiv 2 \pmod{6}\) because both \(14\) and \(2\) leave a remainder of \(2\) when divided by \(6\).
Here are the main points to remember:

• Congruence tells us two numbers behave the same way with respect to a certain modulo.
• The notation \(x \equiv y \pmod{m}\) captures this idea.
• You can think of congruence as a kind of periodicity in numbers where they cycle every \(m\) units.

When solving problems involving congruence, focus on remainders and how numbers revolve around the modulus \(m\). This will help you understand and find solutions to the problems.

The absolute value of a number is a measure of its distance from zero on the number line, regardless of direction.
It is denoted by vertical bars, like this: \(|x|\). For instance, \(|5| = 5\) and \(|-5| = 5\).
In more practical terms, if you have any number, the absolute value simply removes any negative sign and gives you a non-negative result.
Here are the key concepts to remember:

• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is its positive counterpart.
• The absolute value of zero is zero.

When solving problems involving absolute values, focus on understanding what all terms will become once their values are taken as non-negative. This can simplify complicated expressions and make it easier to compare values, especially when looking for the smallest or largest values.

The remainder theorem is a useful tool in polynomial division but is also highly applicable in integer division and modular arithmetic.
Fundamentally, it states that when you divide a number \(a\) by a number \(m\), you can represent \(a\) as the sum of a multiple of \(m\) and a remainder: \(a = km + r\), where \(k\) is an integer and \(0 \leq r < m\).
This remainder \(r\) is what is left over after you divide \(a\) by \(m\).
Here's what you need to know:

• The remainder theorem allows us to express any number as a combination of another number (the divisor) and a remainder.
• It helps us understand how numbers break down into more manageable parts, making calculations simpler.
• The concept of remainders is crucial in modular arithmetic, as it determines how numbers relate to each other within a certain modular system.

Always remember, the remainder when divided by a positive integer \(m\) will always be between \(0\) and \(m-1\), inclusive. This remainder plays a key role in congruence relations where it helps identify equivalent classes of numbers.

Integer division is the division of whole numbers where the result is also a whole number. The division of integers will almost always result in a quotient and a remainder.
For example, if we divide \(17\) by \(5\), we get a quotient of \(3\) and a remainder of \(2\). This can be formally written as \(17 = 3 \times 5 + 2\).
Here are the main points on integer division:

• The quotient is the number of times the divisor fits completely into the dividend.
• The remainder is what is left over after the integer division.
• Integer division is fundamental in modular arithmetic and helps in understanding congruence relations and the remainder theorem.

Always remember, in integer division, the quotient is a whole number and the remainder is a non-negative integer less than the divisor. This simple yet powerful concept is used in various mathematical formulations and proofs.