FACTORS DIVISORS MULTIPLES

Factors

A factor is the part of a multiplication problem that is being multiplied. 

In other words, factor is the number that can go into another number evenly.

For example, when 3 is multiplied by 2 we get the result 6. 

That is,

3 x 2  =  6

Here, both 3 and 2 are factors of 6.

Divisors

A divisor is the number being divided by in a division problem. 

In other words, if a is the divisor of b, then a has to divide b exactly. That is, without remainder. 

For example, 3 is the divisor of 6. Because 3 divides 6 exactly. 

Multiples

Multiple is the result that we get when we multiply a certain number by another one. 

For example, when we multiply 5 by 2, we get 10.

Here 10 is the multiple of both 5 and 2. 

So, the multiples of 5 are 5,10,15,20.

Note : 

If a is the factor or divisor of b, then b is the multiple of a. 

For example, 

3 is the factor or divisor of 6

6 is the multiple of 3 

Practice Problems

Problem 1 : 

Write three multiples of 7.

Solution : 

7 x 2  =  14

7 x 3  =  21

7 x 4  =  28

Problem 2 : 

Find the number of factors of 75600. 

Solution : 

To find the number of factors that 75600 has, first we have to write 75600 in terms of its prime factors. 

Then, we have

75600  =  2⁴ ⋅ 5² ⋅ 3³ ⋅ 7¹

The exponents of the prime factors are

4, 2, 3, 1

To find the number of factors, add 1 to each of the above exponents and multiply them all. 

So, the number of factors of 75600 is 

=  (4 + 1) ⋅ (2 + 1) ⋅ (3 + 1) ⋅ (1 + 1)

=  5 ⋅ 3 ⋅ 4 ⋅ 2

=  120

Note :

These 120 factors of 75600 include 1 and 75600 also. 

Problem 3 : 

Find the number of different factors of 75600. 

Solution : 

In problem no. 2, already we have found the number factors of 75600.

That is 120.

These 120 factors include 1 and 75600 also as factors. 

Because the question targets the number of different factors of 75600, we have to exclude the factor 75600 from those 120 factors. 

Then, the number of different factors of 75600 is 

=  120 - 1

=  119

Problem 4 :

Is 4 the divisor of 152328 ?

Solution :

We already know that if the last two digits of the given number are zeroes or the number formed by the last two digits is a divisible by 4, then the given number is divisible by 4.  

In the given number 152328, the number formed by last two digits is 28 which is divisible by 4.

So, the number 152328 is divisible by 4 exactly. 

4 is the divisor of 152328. 

Problem 5 : 

Is 12 the divisor of 8520 ?

Solution :

We know that if the given number is divisible by both 3 and 4, then it is divisible by 12.

So, let us check whether the given number is divisible by 3.

Sum of the digits :

8 + 5 + 2 + 0  =  15.

Sum of the digits (15) is a multiple of 3.

So, the given number is divisible by 3. 

Now, let us check whether the given number is divisible by 4.

In the given number 8520, the number formed by the last two digits is 20 which is divisible by 4. 

So, the number 8520 is divisible by 4. 

Now, it is clear that the given number 8520 is divisible by both 3 and 4. 

So, the number 8520 is divisible by 12. 

12 is the divisor of 8520.

Problem 6 :

How many factors of the number 2⁸ x 3⁶ x 5⁴ x 10⁵ are multiple of 120 ?

Solution :

If the given number is written as aⁿ x b^m, where a and b are prime numbers, there must be (n + 1)(m + 1) factors will be there.

2⁸ x 3⁶ x 5⁴ x 10⁵ = 2⁸ x 3⁶ x 5⁴ x (2 x 5)⁵

= 2⁸ x 3⁶ x 5⁴ x 2⁵ x 5⁵

= 2⁸⁺⁵ x 3⁶ x 5⁴⁺⁵

= 2¹³ x 3⁶ x 5⁹

Writing down 120 as product of prime factors, we get

120 = 2³ x 3 x 5

When it is multiple of 120, then 120 N

120 N =  2¹³ x 3⁶ x 5⁹

 2³ x 3 x 5 x N = 2¹³ x 3⁶ x 5⁹

N = 2¹⁰ x 3⁵ x 5⁸

(10 + 1) (5 + 1)(8 + 1)

= 11 x 6 x 9

= 594

So, 594 is the answer.

Problem 7 :

How many factors of 1080 are perfect squares?

a) 4         b) 6        c) 8

Solution :

Decomposing 1080 as prime numbers,

1080 = 2 x 2 x 2 x 3 x 3 x 3 x 5

= 2³ x 3³ x 5

= 2² x 3² x 3 x 2 x 5

• 2² is a factor
• 3² is a factor
• 2² x 3² is a factor

For any number 1 is a factor and it is perfect square also, so there are 4 factors for 1080.

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