Finding the Greatest Common Factor (GCF) of 147 and 105 A Step-by-Step Guide

The greatest common factor of 147 and 105 is 7.

Gcf Of 147 And 105

The greatest common factor (GCF) of 147 and 105 can be determined by using various methods. One way to do this is to list all of the multiples of each number, then find the greatest number that they both have in common. Alternatively, find the prime factorization of each number and multiply together the shared factors. The GCF of 147 and 105 is 21. This means that 21 is the largest number that divides evenly into both numbers without leaving any remainder. Knowing the GCF can be useful for simplifying fractions or solving other mathematical equations.

Prime Factorization

Prime factorization is a method used to express any given number as a product of its prime factors. It involves breaking down the number into its smallest prime factors and then writing it as a product of these prime numbers. For example, the prime factorization of 147 is 3 x 7 x 7, while the prime factorization of 105 is 3 x 5 x 7.

Highest Common Factor (HCF)

The highest common factor (HCF) is the largest positive integer that divides two or more numbers without leaving any remainder. In order to find the HCF of two numbers, we must determine all the common factors between them and then choose the largest of these common factors. A common method for finding HCF is to use Prime Factorization. By factoring each number into its prime factors, we can then identify the common factors between them and choose the largest one as our HCF.

Divisibility Tests & Remainders

Divisibility tests are used to determine if one number is completely divisible by another number without having to use long division or other traditional methods of division. In order to determine if one number is divisible by another, we can look at its remainders when divided by that other number. The remainder will tell us if we have a divisible answer or not.

Division Algorithm

The Division Algorithm is a process used to determine how many times one number can be divided evenly into another number without leaving a remainder. This process involves taking two numbers and performing long division on them in order to get an answer that tells us how many times they can be evenly divided without leaving any remainder. It can also be used to find remainders in division by 147 and 105.

Remainders in Division by 147 and 105

When using long division to divide 147 by 105, we get an answer of 1 with a remainder of 42; this means that 105 will go into 147 exactly 1 time with 42 left over as a remainder. When dividing 105 by 147, we get an answer of 0 with a remainder of 105; this means that 147 will not go into 105 at all, leaving us with a remainder of 105 after every attempt at dividing them together.

Euclidean Algorithm for GCD

The Euclidean Algorithm for GCD (greatest common divisor) is an algorithm used to calculate the greatest common divisor between two given numbers. This algorithm works by repeatedly dividing one number by another until either one reaches zero or they reach equal terms (i.e., both equal zero). The greatest common divisor can then be determined from these results.

Explaining Euclidean Algorithm

The Euclidean algorithm works by finding the greatest common divisor between two given numbers by repeatedly dividing them until either one reaches zero or they reach equal terms (both equal zero). By continually subtracting smaller numbers from larger ones, it eventually reaches a point where both are equal – this point being their greatest common divisor – which can then be determined from these results .

Application of Euclidean Algorithm for 147 and 105

In order to apply the Euclidean algorithm for finding GCD between 147 and 105, we first need to subtract smaller numbers from larger ones until both are equal – this point being their greatest common divisor – which can then be determined from these results . To do this, start with 147 and subtract 11 from it so that you have 136; next subtract 9 from 136 so you have 127; now subtract 7 from 127 so you have 120; finally subtract 5 from 120 so you have 115 which equals both 147-11-9-7-5 =105-9-7-5=115 which makes 115 our greatest common divisor between these two numbers .

Multiplicative Inverse Property to Find GCD

Multiplicative inverse property is another method used for finding GCDs between two given integers/numbers A & B such that A*B = GCD(A,B). This property states that if A & B are relatively prime i.e., they share no common factors greater than 1 i .e.. A & B share no major factor , then inverse modulo m exists for both A & B such that m*A + n*B =1 where m & n are integers .

Explaining Multiplicative Inverse Property

The multiplicative inverse property states that if two given integers/numbers A & B are relatively prime i .e.,they share no major factor ,then inverse modulo m exists for both A & B such that m*A + n*B =1 where m & n are integers . This implies that when multiplied together ,the result should give us 1 instead of anything else which indicates they are relatively prime i .e.,they share no major factors greater than 1 when multiplied together resulting in gcd(A,B) =1 making them co-prime since gcd(A,B)=1

Computing GCD Using Multiplicative Inverse Property The multiplicative inverse property allows us to calculate the greatest common denominator between two integers/numbers using an equation: m*A + n*B =1 where m & n are integers . To calculate GCD using this equation ,we must first calculate inverse modulo m for both A&B ,which when multiplied together would give us result as 1 indicating they are relatively prime i .e.,they share no major factor greater than 1 resulting in gcd(A,B)=1 making them co-prime since gcd(A,B)=1 .To calculate gcd using multiplicative inverse property ,we need input values like m&n such as 3&4 respectively for calculating gcd(3&4) which would be calculated as 3*3 + 4 * (-1) = 1 making gcd(3&4 )=3

Gcf Of 147 And 105

GCF or Greatest Common Factor is the largest positive integer that can divide two given numbers without a remainder. In this article, we will be discussing how to find the GCD of two numbers 147 and 105.

Factors/Divisors & Quotients of Numbers

Finding factors/divisors of both the numbers is one of the most important steps in finding their GCD. Factors or divisors are those numbers which can divide the given number without leaving any remainder. In this case, we will be finding factors for both 147 and 105. The factors of 147 are 1, 3, 7, 21, 49 and 147. The factors of 105 are 1, 3, 5, 7, 15 and 105.

Finding quotients during division process is also an important step in finding GCD. Quotients are those numbers which can be divided by another number without leaving any remainder. In this case we will be finding the quotient for 147/105 which is 1 with a remainder of 42.

Deriving GCD Using Prime Factorization Method

Prime factorization method is one of the most popular methods used to find GCD between two numbers. This method involves breaking down both the numbers into their prime factors and then finding their common prime factors to get the GCD between them.
In order to understand prime factorization method better let us take an example:
Let us consider two numbers 147 and 105
The prime factorization of 147 is = 7 x 5 x 3 x 3 = 315 The prime factorization of 105 = 5 x 7 x 3 = 105

Here as you can see both the numbers have one common factor which is 3 and hence their GCD would be equal to 3 only.

Explaining Prime Factorization Method

Prime factorization method involves breaking down both the given number into its prime factors and then finding out their common prime factors to get the GCD between them. The process begins with dividing each given number by its smallest possible prime factor until it becomes completely divisible by that number . After that we use that same divisor for dividing other number as well until it becomes completely divisible by that same number . After repeating this process for each given number ,we finally add up all those common divisors to get our final result which would be our GCD .

Computing GCD Using Prime Factorization Method for 147 and 105

We will take help from our previous example i.e.,147 and 105 ,to compute their GCD using prime factorization method :
The smallest possible prime factor for 147 = 7 (since neither 1 nor 2 divides it) So ,we divide it by 7 147 /7 = 21 So ,we divide 21 by its smallest possible prime factor i..e.,3 21 /3 =7 So ,we divide 7 by its smallest possible prime factor i..e.,7 7 /7=1 Now ,for 105 as well we will follow same procedure as above : The smallest possible prime factor for 105=5 So ,we divide it by 5 105/5=21 So ,now since 21 has already been divided above so we move on to next smallest possible prime factor i..e.,3 21/3=7 Again since 7 has already been divided so we move on to next smallest possible primefactor i..e.,7 7/7=1 Now since all our divisions have resulted in 1 so we stop here . Since all these divisions have resulted in same common Divisor i..e.,3 henceforth our final result would be 3 which would be our Greatest Common Divisor (GCD) between these two numbers i..e.,147 &105 .

FAQ & Answers

Q: What is the GCD of 147 and 105?
A: The greatest common factor (GCD) of 147 and 105 is 21.

Q: How can I find the GCD of two numbers?
A: There are several methods for finding the GCD of two numbers, such as Euclidean Algorithm, Prime Factorization Method, Divisibility Tests & Remainders and Multiplicative Inverse Property.

Q: How do I use the Euclidean Algorithm to find GCD?
A: The Euclidean Algorithm works by repeatedly dividing the larger number by the smaller number until you reach 0. The last non-zero remainder is then used to divide the original larger number, and that answer is the GCD.

Q: What is Prime Factorization Method?
A: Prime Factorization Method is a technique used to determine the prime factors of a given number. To find the GCD using this method, you need to factor both numbers into their prime factors, then multiply together all of their common prime factors. The result will be the GCD.

Q: How do I find Factors/Divisors & Quotients of given numbers?
A: Factors/Divisors are all of the numbers that divide evenly into a given number without leaving a remainder. Quotients are calculated during division process. To find Factors/Divisors & Quotients of given numbers, you can use trial-and-error method or use a divisibility rule to quickly determine if one number divides evenly into another one or not.

The greatest common factor of 147 and 105 is 21. This result can be verified by factoring each number and looking for factors they both share, or by using the prime factorization method. In any case, the GCF of 147 and 105 is 21.