[Maths Class Notes] on Factors of 100 Pdf for Exam

‘A factor is a number that can divide another number completely, without leaving a remainder.’ However,  factors of 100 are the numbers that produce the result as 100 when two numbers are multiplied together.

We know,

1 x 100 = 100 

I.e,1 x 2 x 50 = 100

And,1 x 2 x 2 x 25 = 100

I.e,4 x 25 = 100

Similarly,

20 x 5 = 100

10 x 10 = 100

And,

(-1) x (-100) = 100

I.e,(-1) x 2 x (-50) = 100

and,(-1) x (-2) x (-2) x (-25) = 100

i.e,4 x 25 = 100

Similarly,

(-20) x (-5) = 100

(-10) x (-10) = 100

Here it is clear that 1,-1,2,-2,4,-4,5,-5,25,-25etc are the factors of 100. However, this is not an easy way nor a practical way while dealing with large numbers. Hence, let us learn a few other methods to find the factors.

 

Prime Factorization 

One of the important methods used to find factors of a given number is called prime factorization. Here, upon factorization, we obtain all the prime number factors of the given number. 

Prime numbers are numbers with only 1 and the same number as its factors. For example 2,3,5,7,11,13,etc.

 

Let us now try to apply this to 100 and try to find the prime factorization of 100.

• First, we find out the lowest prime number with which 100 is completely divisible. 

• Since 2 is the lowest prime number that can divide 100 completely, we write 2 on the left-hand side of 100, and the quotient underneath 100 as shown below.

• The lowest prime number with which 50 can be divided completely is also 2 and hence we write 25, the quotient of 50 divided by 2, below 50. 

• When we try to divide 25 by 2, we are obtaining a reminder. Similar is the case with 3. But 25 is completely divisible by 5 and hence we write 25 on the left side and the quotient obtained,5, below 25. 

• Continuing this process, we obtain 2x2x5x5 = 100 the prime factors of 100 are 2 and 5. 

Using these prime factors, we can find more than 100 factors. We can safely assume that the product of these numbers should also be the factors of a hundred. Since,

• 2*2=4; 

• 2*5=10; 10 

• 5*5=25; 25 

• 2*2*5=20;

• 2*5*5=50;

 4,10,20,25, and 50 are also factors of 100. 

We have to note that 1 and the given number are always the factors of the number in question. Hence 1 and 100 are also factors of 100.

 

Factorizing 100 Using Divisibility Rules

Let us try to apply the divisibility rules to 100.

1. 2 is a factor of 100, as 100 is completely divisible by 2.

2. Is 100 divisible by 3? 100/3 yields a remainder of 1; Hence it is not a factor of 3.

3. According to the divisibility rule of 4, if the last two digits of a number are divisible by 4, then the entire number is divisible by 4. Hence 4 is a factor of 100.

4.  If a number ends with 0 or 5, that number is divisible by 5; Hence 5 is a factor of 100.

5. To find out if 100 is divisible by 6, it needs to be divisible by both 2 and 3. As seen earlier,100 is not divisible by 3 and hence 6 is not a factor.

6. Here instead of using the divisibility test of 7, it is easier to try and divide 100 by 7. As this yields a reminder, 100 is not divisible by 7.

7. The divisibility test of 8 warrants us to check whether the last 3 digits of a number are divisible by 8. Hence here we have to divide 100 by 8 to see if 8 is a factor and we obtain a reminder upon doing that. Hence 8 is not a factor of 100.

8. As 3 is a factor of 9 and 3 is not a factor of 100, 9 is also not a factor of 100.

9. As 100 ends in 0,10  is a factor of 100.

Hence, we have obtained all the factors of 100. They are 1,2,4,5,10,20,25,50 and 100.

Using these rules and the prime factorizations, we can find out all the factors of 100. Let us see this using an example. 

 

Example:

We need to check if 27, 38, and 50 are factors of 100.

Step 1. First, let us write the factors of the given numbers.

Step 2. Factors of 27 are 1,3,9,27. We have already seen that 3 and 9 are not factors of 100. Hence 27 is not a factor of 100.

Step 3. Factors of 38 are 1,2,19,38. Here 1 and 2 are factors of 100 but 19 is not a factor of 100. We know that from the prime factorization. Hence, 38 is not a factor of 100.

Factors of 50 are 1,2,5,10,25,50. All of these are factors of 100 and hence 50 is a factor of 100.

 

Types of Factors of 100

The different types of factors are odd, even, and perfect square factors. Let us try to understand them.

The factors that of 100 that are divisible by 2 are even factors and the factors that are not divisible by 2 are called odd factors. Following this, we can see that 2,4,10,20,50 and 100 are even factors and 5 and 25 are odd factors.

The perfect square factors of 100 can be found using prime factorization. We obtained 2*2*5*5=100 from our prime factorization. Hence we have 2 perfect square factors for 100; they are 22 and 52.

We have seen how to obtain factors of 100 and how to check whether a given number
is a factor of 100.  The different types of factors of 100, prime factorization, divisibility rules, etc can help us in determining the factors of large numbers easily and effectively.

Factor Group

A quotient group or factor group is a mathematical group produced by combining related components of a bigger group. Using an equality relation that maintains some of the group organisation. For instance, the cyclic group of addition mod n may be constructed from the group of numbers. Moreover, by finding elements that vary by a multiple of n and designing a group structure. This works on each such class as a separate unit. It is composed of the mathematical discipline known as group theory. In a quotient of a group, the equivalence class of the input is always a normal subset of the original group. And, the remaining equivalence classes are exactly the cosets of that regular subgroup.

 

The reason G/N is termed a quotient group stems from the division of integers. When dividing 12 by 3 one receives the number 4 since one may regroup 12 items into 4 sub-collections of 3 objects. The quotient group is the same approach, however, we end up with a set for a final answer rather than an integer since groups have more organisation than a random collection of items. To explain, when looking at G/N with N a normal subgroup of G, the group framework is utilised to form a natural “regrouping”. But since we began with a group and normal subgroup, the final quotient includes more details than just the number of cosets, but rather it has a group structure itself.