3. Number of factors of given
themes of given number
are:
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Answers
Answer:
Step-by-step explanation:
Let us assume N is a natural number, for which we need to find the factors. If we convert N into the product of prime numbers by prime factorization method, we can represent it as;
N = Xa × Yb × Zc
where X, Y and Z are the prime numbers and a, b and c are their respective powers.
Now, the formula for the total number of factors for a given number is given by;
Total Number of Factors for N = (a+1) (b+1) (c+1)
The formula for the sum of all factors is given by;
Sum of factors of N = [(Xa+1-1)/X-1] × [(Yb+1-1)/Y-1] × [(Zc+1-1)/Z-1]
The formula for the product of all factors is given by;
Product of factors of N = NTotal No. of Factors/2
Example: Find the total number of factors of 90 along with sum and product of all factors.
Solution: Write the prime factorization of 90 first.
90 = 2 × 45 = 2 × 3 × 15 = 2 × 3 × 3 × 5
90 = 21 × 32 × 51
Here, X = 2, Y = 3, Z =5 and a = 1, b = 2, c = 1
Therefore, total number of factors of 90 = (a +1)(b+1)(c+1) = (1+1)(2+1)(1+1) = 2 × 3 × 2 = 12
Sum of factors of 90 = [(21+1-1)/2-1] × [(32+1-1)/3-1] × [(51+1-1)/5-1] = (3/1) × (26/2) × (24/4) = 3 × 13 × 6 = 234
Product of factors of 90 = 90Total factors of 90/2 = 9012/2 = 906
How to Find Factors of a Number?
Knowing how to calculate factors of a number is extremely crucial in maths. The steps to find the factors of a number are given below in a very easy to understand way. An example is taken to make the explanation easier.
Step 1: Choose a number (say, 16)
Step 2: Write the common factors of 16 which will include (16 × 1), (-16 × -1), (8 × 2), (-8 × -2), (4 × 4), and (-4 × -4).
Step 3: Further factor the factors until a prime number is reached. In this case, 8 can be factored further.
Step 4: Write down all the factors again. The (8 × 2) will now become (4 × 2 × 2).
Step 5: Write all the unique number that is obtained.
So, the factors of 16 will be 1, 2, 4, 8, 16, – 1, – 2, – 4, – 8, and – 16. Here, the positive factors of 16 are only 1, 2, 4, 8, and 16.
Another Example:
Consider the number as 40.
40 = 10 × 4
= (5 × 2) × 4
=(5 × 2) × (2 × 2)
= (5 × 2) × (2 × 2)
Now, the factors of 40 will include all the combination from 5 × 2 × 2 × 2 and 1 itself (as 1 × 40 = 40). So, the positive factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. It should be noted that there will also be negative factors whose count have to be even.
How to Calculate Factors of Large Numbers?
To calculate the factors of large numbers, divide the numbers with the least prime number, i.e. 2. If the number is not divisible by 2, move to the next prime numbers, i.e. 3 and so on until 1 is reached. Below is an example to find the factors of a large number.
Example: 1420
Steps Prime Factors Product
Step 1: Divide by 2 2 710
Step 2: Again Divide by 2 2 355
Step 3: Divide by 5 71
In step 3, a prime number is obtained as a product, and so, the process is stopped. The factors will be all the multiples of 1, 2, 2, 5, 71, 355, 710. Now, the positive factors of 1420 will be 1, 2, 4, 5, 10, 20, 71, 142, 284, 355, 710, and 1420.
In the same case, if only prime factors are considered, it is called the prime factorization of that number. In this way, it is easy to factor a number and know its factors and prime factors.
Factors of Some Common Numbers List
Below is a list of common numbers with their factors and prime factors. Each of these links will include the process of factoring of any number along with all the factors, including the prime factors of that number.