Factors and multiples of a number are one of the most commonly used features of numbers in Mathematics. Factors are numbers which we multiply to get another number. Factorization is useful in determination of the common factors among two different numbers. Factors are always found out for numbers which are not prime, or which are composites. This is because prime numbers have two different factors which are predetermined, i.e. the number itself and the number one. However, composite numbers have more than two factors.
Factors can be found out in many ways. For example one can determine the factors of certain small numbers just by looking at it and recalling from the multiplication table like the factors of six can be determined by 3 x 2, which can be found out by recalling the tables of 2 and 3. These are the distinct factors along with the two factors present by default, i.e. the number one and the number six itself. Prime numbers have these two factors only and need no factorisation for finding these.
The next way is as mentioned, factorisation. It is the way of breaking down a composite number into two or more factors using any reducing method. The most common and easy method for small numbers is the pairing method, in which the factors are multiplied in order to complete the base number. The more accurate and versatile method is that of prime factorisation, where the number is divided by the smallest possible prime number until the division gives one or any fractional number in which the number is increased to the next prime number until we get a prime number in the remainder.
Factors of 63 can be found out by two methods respectively; first, we will try by pairing method and then switch to the prime factorisation method. So let us find out what are the factors of 63.
First, we take the base number = 63;
Next we divide it by smallest prime number, i.e. 2 => 63/2 = 31.5,
Since the number we received is a fraction we switch to the next prime number, i.e. 3;
We divide 63 with 3 => 63/ = 21;
Since we get a whole number, we take the whole number and repeat the process with the other primes. After testing with 5 we find that the results are in fractions; hence we proceed with 7;
We divide 63 by 7 => 63/7 = 9;
We received another whole number, but in this case, it is a prime number hence we should abort the process.
This procedure can be demonstrated by the table:
01 x 63 = 63
03 x 21 = 63
07 x 09 = 63
This gives us the factor pairs of 63 are 1, 63; 3, 21; 7, 9;
Hence, we can conclude that all factors of 63 are, 1, 3, 7, 9, 21, and 63 respectively.
Prime Factorisation Method:
This method gives us the prime factors of the composite number
And in this case, we will list all the factors of 63.
First, we take the base number, i.e. 63
Then we divide it by the first feasible prime that is 2;
On dividing with two we get a fraction; hence we cannot consider it as a factor,
Hence, we shift to the next number, i.e. 3;
On dividing with 3 => 63/ 3 = 21;
Hence we get the first factor 3, now we divide the number 21 with primes to get its prime factors, but as we already know that two is not a factor, so we switch to 3;
On dividing 21 by 3; => 21/3 = 7;
Hence, we get another prime factor 3, along with a whole number;
But, in this case, the whole number is a prime itself; therefore, it is considered as a prime factor. Since prime factors do not have extra factors we abort the division process here;
The prime factorisation of 63 can be represented by:
Hence the answer to the question what are all the factors of 63 are 3, 3 and 7; and distinctly the factors of 63 are 3 and 7; this process can also be used to find the common factors of 63.
1. What is the most common method of finding factors?
Solution: The most common method of finding factors of any composite number is by the prime factorisation method. This process is executed by dividing the base number with the prime number in increasing order until we receive a prime number itself or the number one in the remainder. All the quotients for each step are recorded separately, and each of them is individually prime factors of the base number.
2. What is the highest common factor?
Solution: The most important use of the prime factorisation method is to find the highest common factor of two or more numbers. The common factor is that number which can be found in each and every base number when treated with the prime factorisation process. When these common factors are multiplied, they give the greatest common factor. The greatest common factor is a mathematical synonym for the highest common factor. The highest common factor is the measure of the amount or the number of units which can be divided commonly among different quantities of requirement or the base number.
3. How to find Factors of a composite number?
Solution: The process of finding factor can be seen as two different ways, One is the pairing method where pairs of numbers are collected which when multiplied with each other give us the base number, these numbers are called the factors, another method is by prime factorisation and yields us the prime factors of any number in the table format where the divisor block gives us the factors of the numbers