To find the prime factors of 72, we need to look for numbers that can divide 72 exactly without any remainder.
The first step is to divide by the smallest prime number, which is 2.
72 divided by 2 is 36.
Now we need to divide by 2 again.
36 divided by 2 is 18.
We need to keep dividing by 2 until we can’t anymore.
18 divided by 2 is 9.
9 can’t be divided by 2, so we need to try dividing by the next prime number, which is 3.
9 divided by 3 is 3.
Now we have reached a prime number, but we still have to include all the factors that we used to get that number.
So, the prime factors of 72 are 2 x 2 x 2 x 3 x 3.
To check if this is correct, we can multiply all these prime factors together:
2 x 2 x 2 x 3 x 3 = 72
Therefore, we have found all the prime factors of 72.
What is the GCF of 72 using prime factorization?
The GCF (Greatest Common Factor) of 72 can be found using prime factorization. Prime factorization is the method of breaking down a number into its prime factors, prime factors being prime numbers that can only be divided by 1 and themselves.
To find the prime factorization of 72, we can start by dividing it by the smallest prime number, which is 2. We can see that 2 goes into 72 exactly 36 times. Thus, we can write 72 as 2 x 36.
Next, we can look at 36 and find its prime factors by dividing it by 2 again. 2 goes into 36 exactly 18 times. Hence, 36 can be written as 2 x 2 x 18.
Continuing in this way, we can find the prime factorization of 72 as:
72 = 2 x 2 x 2 x 3 x 3
Now that we have the prime factorization of 72, we can find its GCF. The GCF is the greatest factor that is common to two or more numbers, and in this case, we will look for the greatest common factor between 72 and another number.
To find the GCF, we need to compare the prime factors of 72 with those of the other number, and identify the common factors. Let’s say we want to find the GCF of 72 and 120.
The prime factorization of 120 is:
120 = 2 x 2 x 2 x 3 x 5
Looking at the prime factorizations of both 72 and 120, we can see that they have 2, 2, and 3 as common factors. Therefore, the GCF of 72 and 120 is:
GCF(72, 120) = 2 x 2 x 3 = 12
So, the GCF of 72 using prime factorization is 2 x 2 x 2 x 3 = 12.
How do you find prime factors?
Prime factorization is the process of decomposing a composite number into its prime factors. To find the prime factors of a number, one needs to follow the step-by-step process:
1. Start with the number that needs to be factored and write it down.
2. Find the smallest prime factor of the number. If the number is itself a prime number, the factor is the number itself.
3. Divide the number by the smallest prime factor found in step 2.
4. Write both the divisor and quotient below the original number.
5. Repeat Steps 2 through 4 with the quotient obtained in Step 4 until no further division is possible.
6. The prime factors of the original number are the set of prime numbers obtained from the above steps.
For example, let’s find the prime factors of the number 84.
1. Start with the number 84.
2. The smallest prime factor of 84 is 2.
3. Divide 84 by 2 to get 42.
4. Write 2 and 42 below 84.
5. The smallest prime factor of 42 is 2.
6. Divide 42 by 2 to get 21.
7. Write 2 and 21 below 42.
8. The smallest prime factor of 21 is 3.
9. Divide 21 by 3 to get 7.
10. Write 3 and 7 below 21.
11. 7 is a prime number, so we stop.
12. The prime factors of 84 are 2, 2, 3, and 7.
Finding prime factors is not difficult, but it requires following the above steps carefully. Prime factorization is useful in many areas such as cryptography, number theory, and computer science.
What is the factor tree of prime factorization of 72 using exponents?
The prime factorization of 72 can be represented using exponents in the following factor tree:
72
/ \
2^3 3^2
/ \
2 2
To find the prime factorization of 72, we begin by dividing it by its smallest prime factor, which is 2. We get 2 x 36. We then continue to factor 36 by dividing it by 2 again, resulting in 2 x 18. We repeat this process until we obtain only prime factors. The factor tree illustrates this process, showing that 72 can be expressed as the product of 2 raised to the power of 3 and 3 raised to the power of 2, i.e., 2^3 x 3^2.
Using exponents in the prime factorization of 72 allows us to represent the same information in a more compact way. Instead of writing out the full factor tree or using multiplication symbols, we can simply write 2^3 x 3^2 to represent the prime factors of 72. This is a useful shorthand when dealing with larger numbers, as it allows us to express their prime factorization in a way that is easier to read and work with.
understanding the prime factorization of a number and how to express it using exponents is an important skill in mathematics and can be applied in many different contexts.
What is the HCF of 72 and 64 by listing method or prime factors method?
The HCF (Highest Common Factor) of 72 and 64 can be found using two methods: listing method or prime factors method. Let’s take a look at both methods.
Listing Method:
To find the HCF of 72 and 64 using the listing method, we need to list all the factors of both numbers and then find the highest common factor from those lists.
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Factors of 64: 1, 2, 4, 8, 16, 32, 64
From the above lists, we can see that the highest common factor is 8. Therefore, the HCF of 72 and 64 using the listing method is 8.
Prime Factors Method:
To find the HCF of 72 and 64 using the prime factors method, we need to express both numbers as products of their respective prime factors and then find the highest common factor from those lists.
Prime factors of 72: 2 x 2 x 2 x 3 x 3
Prime factors of 64: 2 x 2 x 2 x 2 x 2 x 2
From the above lists, we can see that the common prime factors are 2 x 2 x 2. Therefore, the HCF of 72 and 64 using the prime factors method is 8.
In both methods, we got the same answer i.e. 8. Therefore, the HCF of 72 and 64 is 8.
What method is used to find the prime factors of a number?
The method used to find the prime factors of a number is called prime factorization. Prime factorization is the process of breaking down a number into its prime factors, or factors that are only divisible by 1 and themselves. To find the prime factors of a number, you first need to determine if the number is itself a prime number.
If the number is prime, it cannot be broken down any further, and the only prime factor will be the number itself.
However, if the number is not prime, you can begin the process of prime factorization by dividing the number by the smallest prime number that is a factor of the original number, which is usually 2. Continue to divide the result by the smallest prime factor until you are left with only prime factors.
This process can be done using a factor tree, which helps visualize the prime factors and their corresponding non-prime factors.
For example, to find the prime factors of the number 60, you would begin by dividing 60 by 2, which equals 30. 30 can be divided by 2 again to equal 15, and 15 can be further divided by 3 to equal 5 – all of these numbers are prime factors. Therefore, the prime factors of 60 are 2 x 2 x 3 x 5, which can be written in exponential form as 2^2 x 3 x 5.
Prime factorization is an important tool in mathematics and has many practical applications, including in cryptography and computer security. It allows us to better understand the structure of numbers and can be used to simplify and solve complex equations involving factors.