How to write the fraction 28/40 in the simplest form

Simplifying the Fraction

To simplify the fraction $\frac{28}{40}$ to its simplest form, we need to find the greatest common divisor (GCD) of the numerator and the denominator.

Finding the GCD

The GCD is the largest number that divides both the numerator (28) and the denominator (40) without leaving a remainder. We first list the factors of each number:

• Factors of 28: $1, 2, 4, 7, 14, 28$
• Factors of 40: $1, 2, 4, 5, 8, 10, 20, 40$

The common factors are 1, 2, and 4. Among these, the largest factor is 4.

Simplifying the Fraction

We divide both the numerator and the denominator by their GCD:

Thus, the fraction $\frac{28}{40}$ in its simplest form is:

Definition

The greatest common divisor (gcd) of two integers is the largest positive integer that divides both numbers without leaving a remainder. It is a key concept in number theory and is used in various mathematical computations such as simplifying fractions.

Properties

1. Commutative Property:

   $$\text{gcd}(a, b) = \text{gcd}(b, a)$$

2. Associative Property:

   $$\text{gcd}(a, \text{gcd}(b, c)) = \text{gcd}(\text{gcd}(a, b), c)$$

3. Non-negativity: The gcd of two numbers is always a non-negative integer.

4. Identity Property: The gcd of any number and 0 is the number itself,

   $$\text{gcd}(a, 0) = a$$

Euclidean Algorithm

The Euclidean Algorithm is an efficient method for computing the gcd of two numbers. It is based on the principle that the gcd of two numbers also divides their difference.

1. Given two numbers $a$ and $b$ where $a > b$, the algorithm proceeds as follows:

2. Find the remainder $r$ when $a$ is divided by $b$:

   $$r = a \mod b$$

3. Replace $a$ with $b$ and $b$ with $r$:

   $$a \leftarrow b, \quad b \leftarrow r$$

4. Repeat the process until $b$ becomes 0. At this point, $a$ will be the gcd of the original two numbers.

Example Calculation

Let's find the gcd of 48 and 18 using the Euclidean Algorithm:

1. $48 \div 18 = 2$ remainder $12$

   $$48 = 18 \cdot 2 + 12$$

2. $18 \div 12 = 1$ remainder $6$

   $$18 = 12 \cdot 1 + 6$$

3. $12 \div 6 = 2$ remainder $0$

   $$12 = 6 \cdot 2 + 0$$

Since the remainder is now 0, we stop, and the gcd is the last non-zero remainder:

Applications

• Simplifying Fractions: The gcd is used to reduce fractions to their simplest form.
• Cryptography: Algorithms like RSA rely on the gcd for key generation.
• Algebraic Structures: In ring theory and module theory, gcds help in understanding the structure of rings and modules.

Understanding and computing the greatest common divisor is foundational for many areas of mathematics and computer science.

Prime Factorization

Prime factorization is the process of breaking down a composite number into a product of its prime factors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Steps for Prime Factorization

1. Divide the number by the smallest prime number (2, 3, 5, etc.).
2. Continue dividing by prime numbers until the quotient is a prime number.
3. Express the number as a product of these prime factors.

Example

Let's take the number 60 and find its prime factors:

1. Start with the smallest prime number (2):

   $$\frac{60}{2} = 30$$

2. Continue dividing by 2 as long as possible:

   $$\frac{30}{2} = 15$$

3. 15 is not divisible by 2, so move to the next smallest prime number (3):

   $$\frac{15}{3} = 5$$

4. 5 is a prime number, so we stop here.

Thus, the prime factorization of 60 is:

We can also write this as:

Importance

Prime factorization has several important applications in mathematics, including:

• Simplifying fractions
• Finding the greatest common divisor (GCD)
• Finding the least common multiple (LCM)
• Cryptography and number theory

Understanding this concept is fundamental for solving various mathematical problems efficiently.