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Chapter 5: Problem 3

Write a congruence to say that \(x=12 k+5\) for some integer \(k\).

Short Answer

Expert verified
The congruence for the equation \(x = 12k + 5\) is \(x \equiv 5 \pmod{12}\).

Step by step solution

01

Express the equation in the congruence form

To write the equation, x = 12k + 5, in the form of a congruence, we simply rewrite it as: x 鈮� 5 (mod 12) By doing this, we are stating that the difference between x and 5 is divisible by 12. Essentially, this means that x has a remainder of 5 when divided by 12, which is the same as saying that x = 12k + 5 where k is an integer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Congruence in Modular Arithmetic
In modular arithmetic, congruence is an expression that shows how two numbers can relate to each other within a certain modulus. Imagine that congruence is like a clock that resets after reaching a certain number, called the modulus. When we say \( x \equiv 5 \pmod{12} \), it tells us that when \( x \) is divided by 12, the remainder is 5.
  • This means \( x \) and 5 are in the same position on a clock with 12 hours.
  • Congruence allows multiple numbers to share this special relationship, such as 17 and 5 when considered under modulus 12.
Remember, congruence joins numbers with similar remainders in modular arithmetic, making calculations easier and more systematic.
Understanding Integers
Integers are whole numbers that make up the set of negative numbers, zero, and positive numbers. So, when we say "integer," we mean numbers like -2, 0, 1, and beyond in both directions. They're important in describing congruences because they help determine the values we work with.
  • In our problem, \( k \) is an integer, which means it can take any whole number value.
  • Being comfortable with different integer values helps in exploring all possibilities in equations and congruences.
Every integer has a place in modular equations, since it contributes to finding numbers that fit a specific remainder when divided.
The Role of Divisibility
Divisibility tells us whether one integer can be divided by another without leaving a remainder. In the expression \( x = 12k + 5 \), divisibility plays a critical role.
  • We are investigating when \( x \) is divisible by 12 with a remainder involved.
  • If \( x \equiv 5 \pmod{12} \), this shows \( x - 5 \) is divisible by 12.
Divisibility in congruence ensures we can identify numbers that perfectly fit into a cycle or sequence within a modulus, making solutions clear and straightforward.
Understanding Remainders
Remainders are what is left over after division. They are essential in modular arithmetic because they define congruences. When you divide one number by another, the remainder is what's left when the divisor cannot completely divide the number.
  • For \( x = 12k + 5 \), the number 5 is the remainder when \( x \) is divided by 12.
  • Remainders tell us how far apart numbers are within a modulus sequence.
By focusing on remainders, we can predict and analyze patterns in numbers, making it easier to handle complex equations and number relationships.

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