Cryptography is a method of protecting information and communicating cryptography through the use of codes. Prime factorization plays an important role for the coders who want to create a unique code using numbers that is not too heavy for computers to store or process quickly. For this, we first find the prime factorization of both the numbers.
Next, we consider the following:. Solution: We first find the prime factorizations of both the numbers. The prime factorization of is shown below:. HCF is the product of the smallest power of each common prime factor. LCM is the product of the greatest power of each common prime factor.
Example 1: Can you help Charlize to express as the product of prime factors? Also, can you tell if this factorization is unique? Example 2: Jenifer was given a task by her teacher to find the lowest common multiple of 48 and 72 using prime factorization. Can you help her? We will find the prime factorizations of 48 and The prime factorization of 48 is shown below:.
The LCM or lowest common multiple of any 2 numbers is the product of the greatest power of the common prime factors. Example 3: Jane has to prove that the prime factorization of 40 will always remain the same. She is confused, help her prove it. Jane can use the division method and factor tree method to prove that the prime factorization of 40 will always remain the same.
Jane knows that 40 can be factored as 5 and 8. The composite number 8 can further be broken down as a product of 2 and 4. All this because they are an irreducible part of the very fabric of the universe. By subscribing, you agree to our Terms of Use and Privacy Policy. You may unsubscribe at any time. By Zachary Tomlinson. Follow Us on. Sponsored Stories. If you continue to use this site, you consent to our use of cookies. Stay on top of the latest engineering news. A common multiple of two numbers is a number that is a multiple of both numbers.
Suppose we want to find common multiples of and We can list the first several multiples of each number. Then we look for multiples that are common to both lists—these are the common multiples. We see that and appear in both lists. They are common multiples of and We would find more common multiples if we continued the list of multiples for each. The smallest number that is a multiple of two numbers is called the least common multiple LCM. So the least LCM of and is.
Find the LCM of and by listing multiples. List the first several multiples of and of Identify the first common multiple. The smallest number to appear on both lists is so is the least common multiple of and. Notice that is on both lists, too. It is a common multiple, but it is not the least common multiple. Find the least common multiple LCM of the given numbers:.
Another way to find the least common multiple of two numbers is to use their prime factors. Then we write each number as a product of primes, matching primes vertically when possible. Notice that the prime factors of and the prime factors of are included in the LCM. By matching up the common primes, each common prime factor is used only once.
This ensures that is the least common multiple. Find the LCM of and using the prime factors method. Find the LCM using the prime factors method. Find the LCM using the prime factors method:. Find the Prime Factorization of a Composite Number. In the following exercises, find the prime factorization of each number using the factor tree method.
In the following exercises, find the prime factorization of each number using any method. In the following exercises, find the least common multiple LCM by listing multiples. In the following exercises, find the least common multiple LCM by using the prime factors method.
In the following exercises, find the least common multiple LCM using any method. Grocery shopping Hot dogs are sold in packages of ten, but hot dog buns come in packs of eight. What is the smallest number of hot dogs and buns that can be purchased if you want to have the same number of hot dogs and buns? Hint: it is the LCM! Grocery shopping Paper plates are sold in packages of and party cups come in packs of What is the smallest number of plates and cups you can purchase if you want to have the same number of each?
Do you prefer to find the prime factorization of a composite number by using the factor tree method or the ladder method? Do you prefer to find the LCM by listing multiples or by using the prime factors method? Why or why not? Simplify Expressions Using the Order of Operations. In the following exercises, simplify the following expressions by combining like terms.
Translate English Phrases to Algebraic Expressions. In the following exercises, translate the following phrases into algebraic expressions. Jack bought a sandwich and a coffee. The cost of the sandwich was more than the cost of the coffee. Call the cost of the coffee Write an expression for the cost of the sandwich. Call the number of novels Write an expression for the number of poetry books. Determine Whether a Number is a Solution of an Equation.
In the following exercises, determine whether each number is a solution to the equation. So the first thing we have to worry about is what is even a prime number? And just as a refresher, a prime number is a number that's only divisible by itself and one, so examples of prime numbers-- let me write some numbers down.
Prime, not prime. So 2 is a prime number. It's only divisible by 1 and 2. Now, 4 is not prime, because this is divisible by 1, 2 and 4. We could keep going. I think you get the general idea. You move to 7, 7 is prime. It's only divisible by 1 and 7. Prime is not the same thing as odd numbers.
Then if you move to 10, 10 is also not prime, divisible by 2 and 5. And we could keep going on like this. People have written computer programs looking for the highest prime and all of that. So now that we know what a prime is, a prime factorization is breaking up a number, like 75, into a product of prime numbers.
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