Here we are going to discuss, what are the factors of 99 and how many factors 99 has and what are the factors of 99 in pairs . Here are some methods, showing how to find the factors of 99.
Actually, Factor is a number which divides any number completely without leaving remainder.
Or simply we can say that if we multiply two whole numbers (positive & negative) and it gives a product. Then the numbers that we multiply, are the factors of the product.
For example, if we multiply 9 to 11, it gives 99. Here 9 and 11 are positive factors of 99 or we multiply -3 to -33, it gives 99. So -3 and -33 are negative factors of 99.
What are the Factors of 99 ?
Since factors of 99 are the numbers, which divides 99 completely without leaving remainder. So here we are
going to divide 99 by all numbers up to 99. Let’s see the process –
We have to stop the process, if any number of column ‘B’ appears in column ‘A’.
Here we stop the division when 11 appears in column ‘A’, which has already been came in column ‘B’.
Hence, all the factors of 99 in above table are : 1, 3, 9, 11, 33, 99
Similarly, negative factors of 99 are : -1, -3, -9, -11, -33, -99
Take another example, Factors of 25
In above example, no element of Column ‘B’ appears in Column ‘A’, but we stopped the process
when 5 comes, because on dividing by 5, it produces the same number (5 = 5) .
So we have to stop the process when any element comes in both columns at same level.
And this element will consider as a factor.
Note :- This situation occurs with a perfect square number, such as 9, 16, 25, 36, 49 etc.
Factor pairs of 99
As we saw in above table, 1 multiply by 99 gives 99. So (1, 99) or (99, 1) is a factor pair of 99.
Similarly, (3, 33) and (9, 11) are another factor pairs of 99.
In example of 25 ; (1, 25) or (25, 1) and (5, 5) are factor pairs.
Methods of Prime Factors of 99
Here we will discuss, mainly two methods of factorization to find Prime Factors of any number.
Method #1 (Division Method)
Step 1 : Divide the number by the smallest prime number, which divides the number exactly.
Step 2 : Divide the quotient again by the smallest prime number. If quotient is not exactly divisible by the
smallest prime number, choose next smallest prime number.
Step 3 : Repeat Step 2 again and again till the quotient becomes 1.
Step 4 : All the prime numbers used in above process are the prime factors.
Check : Multiply all the prime factors, the multiplication should be the number itself.
Caution : In this method, only prime numbers will be use to divide.
Take example of 99 and 20
In above example of 99
Step 1 : First of all, we divide 99 by smallest prime number i.e. 2, But 2 doesn’t divide 99 completely. Try next prime number 3.
(Note – if remainder will not zero, means doesn’t divide exactly, try next prime number to divide)
Step 2: This time we got 33 as quotient without leaving any remainder.
Step 3: Again we divide 33 by prime number i.e. 3, it gives 11 as quotient.
Step 4: Since 11 is a prime number, hence it will divide by itself only.
Hence, 99 = 3 x 3 x 11. So 99 has two prime factor i.e. 3 and 11.
In second example of 20
Step 1 : First of all, we divide 20 by smallest prime number i.e. 2 and we got 10 as quotient and remainder is zero. (Note – if remainder will not zero, means doesn’t divide exactly, try next prime number to divide)
Step 2: Again we divide the quotient 10 by smallest prime number i.e. 2. We got 5 as quotient.
Step 3 : Since 5 is a prime number, hence it will divide by itself only.
Hence, 20 = 2 x 2 x 5 = 22 x 5
So 20 has two prime factor i.e. 2 & 5.
Method #2 (Factor Tree Method)
Step 1 : Write a number as root of the tree
Step 2 : Find the factors of the number, each factor will be considered as root again (now, we got 2 new root)
Step 3 : Repeat step 2 until we can’t factor any more (or until we got prime factor)
Step 4 : The end nodes are the prime factors.
Factor Tree of 99
In First Tree ( see above figure), We can initiate the tree by choosing any two factors. Here we choose 3 and 33 as factors of 99.
Further, we divide 33 in two factors, i.e. 3 and 11. Since 3 and 11 are prime numbers and can not be further split.
Hence, we got 3, 3, 11 are nodes, in which 3 and 11 are unique nodes.
Similarly, in other factor trees, we can initiate by any two factors on Main Node (i.e. 99). And apply above process till we got prime nodes.
Note:- The Factor Tree may be made so many types depends upon choosing the initial and middle factors.
In second tree, we choose 11 and 9 as initial factors of 99. Since 11 is a prime number and can’t be split further. And 9 can be split into 3 and 3.
So in this tree 11, 3, 3 are nodes. and 3 and 11 are unique nodes and these are prime factors.
All nodes and unique nodes will be same, it doesn’t depend tree. We can choose any initial nodes and middle nodes.
Number of factors of 99
To find the number of factors, we use Method #1 (Division Method).
In our example, as we saw above 99 = 3 x 3 x 11 = 32 x 11 means 3 to the power 2 multiply 11 to the power 1
Number of factors is = (power + 1)(power + 1)(power + 1)………. = (2 + 1)(1 + 1) = 3 x 2 = 6
if we consider both positive and negative factors, then
total no. of factors = 6 (positive) + 6 (negative) = 12
Take another example, 20 = 2 x 2 x 5 = 22 x 5
Number of factors of 20 is = (2 + 1)(1 + 1) = 3 x 2 = 6
and if we consider both positive and negative factors, then
total no. of factors = 6 (positive) + 6 (negative) = 12
I hope, this step-by-step tutorial will be helpful to understand factorization method of 99.
Try another number 40 for factors.
Take a another look of factorization on Quora.