Question

Prove that two odd integers whose difference is 32 are coprime.

Short answer

Yes, two odd integers with a difference of 32 are coprime.

Step-by-step solution

Define odd integers

To prove that two odd integers whose difference is 32 are coprime, start by defining the integers. Let the two odd integers be \(a\) and \(b\). Since they are odd, we can express them as \(a = 2m + 1\) and \(b = 2n + 1\) for some integers \(m\) and \(n\).

Express the difference

Calculate the difference between the two integers: \(a - b = (2m + 1) - (2n + 1) = 2m + 1 - 2n - 1 = 2(m - n)\). We know from the problem statement that this difference is 32, so we set this equal to 32 and solve: \(2(m - n) = 32\).

Solve for m-n

Divide both sides of the equation by 2 to find \(m - n\): \(m - n = \frac{32}{2} = 16\). This means \(m - n = 16\).

Express integers in terms of m-n

Express \(a\) and \(b\) to see if they have a common factor. Since \(m - n = 16\), express \(m\) as \(n + 16\). Substitute back into the expressions for \(a\) and \(b\) to get: \(a = 2(n + 16) + 1 = 2n + 32 + 1 = 2n + 33\) and \(b = 2n + 1\).

Check for coprime condition

To be coprime, integers \(a\) and \(b\) must have a greatest common divisor (gcd) of 1. Calculate the gcd of \((2n + 33)\) and \((2n + 1)\). The difference is \(32\), as expressed already, which does not have any common factors with the integers themselves other than 1, confirming they are coprime.

Key concepts

Odd Integers

Odd integers are numbers that cannot be evenly divided by 2. This means when you divide an odd number by 2, you will always have a remainder of 1. They are part of the number line just like even integers. Even numbers are those that can be perfectly divided by 2 without any remainder.
We can represent any odd integer using the formula:

• \( a = 2m + 1 \)

where \(m\) is any integer. This formula simply states that if we take any whole number \(m\), multiply it by 2 (which makes it even), and then add 1, the result will always be an odd number.
Examples of odd numbers are 1, 3, 5, 7, and so on. In fact, the sequence of odd numbers grows by adding 2 (the difference between consecutive odd numbers is 2). Understanding the basic structure of odd integers is essential, especially when used in various number theory problems.

Coprime Integers

Coprime integers, also known as relatively prime integers, are two or more numbers that have no common positive divisors other than 1. This means their greatest common divisor (GCD) is 1. If two numbers share any other common factors, they are not coprime.
To find out if two numbers are coprime, you would use the following steps:

• List the factors of each number.
• Identify any common factors.
• If the only common factor is 1, then the numbers are coprime.

For example, consider the numbers 9 and 28:

• Factors of 9: 1, 3, 9
• Factors of 28: 1, 2, 4, 7, 14, 28
• The only common factor is 1, so 9 and 28 are coprime.

This is a fundamental concept in number theory because coprimeness often shows up in problems related to divisibility, prime numbers, and greatest common divisors.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. It's a key concept used to simplify fractions and solve divisibility problems.
For example, consider the numbers 54 and 24:

• Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Common factors: 1, 2, 3, 6
• The GCD is 6.

Finding the GCD can be done using several methods like listing factors or using the Euclidean algorithm. The latter is often preferred for larger numbers as it involves fewer calculations. Understanding the GCD is essential in math as it helps in reducing fractions and simplifying algebraic expressions.