In mathematical terms, factoring or factorization of a number involves writing that number as a product of many other numbers. These factors are usually simpler and smaller objects of the same kind. In other words, factorization is the mathematical decomposition of a specific number.
For example, when we factorize the number 15, we get 3 * 5. Here, we will learn all about what are the factors of 17 and look into some of the properties of the factors of 17. But before that, let us touch upon some basics.
[Image will be Uploaded Soon]
Factors are those whole numbers (0, 1, 2, 3, ….) that divide the given number exactly and completely without leaving any remainder.
Numbers that have only two factors, one of them includes the number itself, and the other one is number 1, are prime numbers. Some of the prime numbers are 5, 7, 2, 11, etc.
They are in contrast to prime numbers in the sense they have more than 2 factors. Whole numbers that are divisible by more than 2 numbers are composite numbers, for example, 18 = 2 * 3 * 3
From the above definitions, 17 is a prime number. To prove that 17 is a prime number, it should be divisible by at least one prime number which is less than or equal to √17 ≈ 4.12. But 17 cannot be divided evenly by either 2 or 3 hence we infer that it is a prime number.
To find all the factors of 17, we need to divide 17 by all the whole numbers starting from 1 up to 17, which can give us an even quotient. When we do that we find out: 17/1 = 17 (as it gives a remainder of 0) and 17//7 = 1 (Here also the remainder is 0). So all the divisors of these equations are the factors of 17 which are 1 and 17 (Positive factors of 17)
All the positive factors of 17 can be converted into its negative factors. Hens the list of negative factors of 17 is -1 and -17.
Since we just saw that 17 has two positive factors and two negative factors hence 17 has a total of 4 factors.
In number theory, we define prime factors of a positive integer as the numbers that exactly divide the integer. This is given by representing 17 due to the product of prime numbers hence the prime factorization of 17 is 17. 17 is not a composite number as none of the two factors of 17 (1 and 17) is the same number.
Factor pairs of a number are two numbers which when multiplied with each other give that number as a result.
All the positive factors of 17 are detailed below:
1 * 17 = 17 and 17 * 1 = 17 hence pair factors of 17 are (1, 17) and (17, 1)
17 also has negative factors since minus multiplied by minus gives a positive number. All the negative factors of 17 are detailed below:
-1 * -17 = 17 and -17 * -1 = 17 hence pair factors of 17 are (-1, -17) and (-17,-1)
Below are a few important properties of factors:
Every whole number can be expressed as the product of number 1 and itself, i.e. each number is a factor of itself.
1 is the factor of every number, i.e. all numbers are divisible by 1.
Every number is a factor of 0, i.e. all numbers when multiplied with 0 give 0 as a result.
The smallest factor of a multiple is 1, and the greatest factor of a multiple is the multiple itself.
All numbers (except 1) have at least 2 factors which are the number 1 and itself.
Q1. Explain the Concept of a Factor Tree.
Ans: In a factor tree, the possible factors of a number (along with their subfactors) are represented graphically in a tree format. A factor tree helps in simplifying factorization. To begin creating the tree, first, we list down factors of a number then keep breaking factors of factors recursively till we reach the terminal leaf nodes. These terminal nodes are basically prime factors which is also the prime factorization of the original whole number. When you construct the tree, we must consider and remember the second item in the factor tree. An example of such a tree is shown below:
[Image will be Uploaded Soon]
Q2. What are the Divisibility Rules?
Ans: Divisibility means how a given integer number is divisible by a given divisor. This rule is a shorthand system that helps determine what is divisible and what is not, even without doing too many calculations. In this rule, we have rules for even number and odd number factors. Here are the various divisibility rules:
All integers (not fractions) are divisible by 1
For a number to be divisible by 2 the last digit must be an even number. For example, 262 has an even number 2 at the end; hence it is divisible by 2 but 263 is not.
For a number to be divisible by 3 the sum of its digits should be divisible by 3 for example 12 = 1 + 1 = 3 and as such 3 is divisible by 3 hence 12 is also divisible by 3.
A number is perfectly divisible by 4 when the last two digits must be divisible by 4. For example, 1312 has 12 at the end, and 12 is divisible by 4 (12/4 = 3) hence 1312 is divisible by 4.
Any number with their last digits as 5 or 0 is divisible by 5.
If a number passes both the divisibility rules of numbers 2 and 3 stated above, then it is divisible by 6. For example, 114 -> It is an even number since it has 4 at the end and its sum is 6, which is divisible by 3 hence 114 is divisible by 6.
A number will be divisible 7 if you double the last digit of that number and then subtract it from the remaining digits of the number then the result should be divisible by 7. For example, in number 672, 2 is the last digit whose square is 4, so we subtract 4 from the remaining digits which are 67, 67 - 4 = 63, and 63 is divisible by 7 (63/7 = 9) so 672 is divisible by 7.
If the last 3 digits of a number are divisible by 8 then that number is divisible by 8, for example, 109816, here 816/8 = 102 hence 109816 is divisible by 8.
For a number to be divisible by 9 , if you add up the digits of that number it must be divisible by 9. For example 18 = 1 + 8 = 9 and 9 is divisible by 9 (9/9 = 1) hence 18 is also divisible by 9.
Any number which ends in a 0 is divisible by 10, for example, 78950.