Find the prime factorisation of each of the following numbers by division method
Answers
Answer:
Step-by-step explanation:
ne method for producing the prime factorization of a natural number is to use what is called a factor tree.
The first step in making a factor tree is to find a pair of factors whose product is the number that we are factoring. These two factors are the first branching in the factor tree. There are often several different pairs of factors that we could choose to begin the process. The choice does not matter; we may begin with any two factors. We repeat the process with each factor until each branch of the tree ends in a prime. Then the prime factorization is complete. The Fundamental Theorem of Arithmetictext annotation indicator guarantees that all prime factorizations of the same number will result in the same, unique prime factorization for the number.
Example: We show two of the ways of constructing a factor tree for 24.
Trees1.PNG
Continue factoring each tree until complete.
Trees2.PNG
Note that each tree ends with the unique prime factorization of 24 = 2 · 2 · 2 · 3 = 23 · 3.
There are two different styles for writing down the factor tree of a natural number. In the first style, as soon as we obtain a prime number in one of the branches, we circle it and then do not work on that branch any more. If a number at the end of a branch is still not prime (a composite), we find two factors for that value. Continue this process until the value at the end of each branch is a circled prime number. The prime factorization is the product of the circled primes.
Example:
Trees3.PNG
So the prime factorization of 24 is 24 = 2 · 2 · 2 · 3 = 23 · 3.
A good way to check the result is to multiply it out and make sure the product is 24.
For the other style of factor tree, we maintain the product of the original value at each level of the factor tree by extending the branch (bringing down) for any prime obtained on the way to getting all of the branches to end in prime numbers.The following example shows this style and also how we may start with a different pair of factors and still come out with the same prime factorization for the natural number.
Example:
Trees4.PNG
Some people prefer this method because each level still multiplies to be the original number, and by bringing down the primes, we are less likely to miss them and leave them out of our prime factorization.
Notice that the prime factorization is still 24 = 2 · 2 · 2 · 3 = 23 · 3 even though we started with 2 · 12 instead of 6 · 4.
Self-Check Problem
Make another factor tree for 24 where two different factors are used in the first step than were used above.
Solutiontext annotation indicator
Make a factor tree to find the prime factorization for 36.
Solutiontext annotation indicator