FACTORS AND DIVISORS WORKSHEET

1)  Write three multiples of 12.

2)  Find the number of factors of 75600. 

3)  Find the number of different factors of 75600. 

4)  Is 4 the divisor of 5698328 ?

5)  Is 12 the divisor of 8520 ?

6)  The expression 5⁴ + 7³ is divisible by which of the following numbers ?

a)  2      b)  5      c)  10      d)  12

7)  444²²² + 555¹¹¹ is divisible by which of the following numbers ?

a)  2  but not 5      b)  3 but not 5 and 2

c)  2 and 5 but not 3       d) 2, 3 and 5

8)  Let N be the least positive multiple of 11 that leaves a remainder of 5, when divided by 6, 12, 15 and 18. Which of the following is correct ?

a)  900 < N < 1000       b)  1000 < N < 1100

c)  1100 < N < 1200       d) 1200 < N < 1300

Detailed Answer Key

1. Answer : 

12 x 2  =  24

12 x 3  =  36

12 x 4  =  48

2. Answer : 

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. 

3. Answer : 

In the above answer, 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

4. Answer :

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 5698328, the number formed by last two digits is 28 which is divisible by 4.

So, the number 5698328 is divisible by 4 exactly. 

4 is the divisor of 5698328. 

5. Answer :

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.

6. Answer :

Find the value of 5⁴ + 7³ :

5⁴ = 625

7³ = 343

5⁴ + 7³ = 625 + 343

= 968

Divisibility test for 2 :

Since it is even number, it is divisible by 2.

Divisibility test for 3 :

9 + 6 + 8 = 23

Since the sum of the digits is not divisible by 3, 5⁴ + 7³ is not divisible by 3.

Divisibility test for 5 :

968 does not ends with 0 or 5, it is not divisible by 5.

It is divisible by 2.

7. Answer :

= 444²²² + 555¹¹¹

Divisibility test for 2 :

• In 444²²², the product of even by another even will produce even number. 
• In 555¹¹¹, the product of odd by another odd number will produce odd number.

Sum of odd and even will be equal to odd. Then, it is not divisible by 2.

Divisibility test for 3 :

• 444 = 4 + 4 + 4 ==> 12, which is divisible by 3.
• 555 = 5 + 5 + 5 ==> 15, which is divisible by 3

Since these numbers are divisible by 3, its sum will also be divisible by 3.

Divisibility test for 5 :

A number ends with 0 or 5 will be divisible by 5.

• 444 ends with 4, it is not divisible by 5.
• 555 ends with 5, it is divisible by 5.

Then, the sum will not be divisible 5. So, the answer is 

3 but not 5 and 2

8. Answer :

N is a number which is multiple of 11 and leaves the remainder 5.

Finding the common number for 6, 12, 15 and 18.

lcm = 2 x 3 x 2 x 5 x 3

= 180

The required number must be a multiple of 180 added with 5.

Required number = 180 N + 5

It is divisible by 11.

When N = 5

Required number = 180(5) + 5 ==> 905

Divisibility test for 11 :

9 + 5 = 14

Difference between sum of the values in the odd place - even places

= 14 - 0

= 14

When N = 7

Required number = 180(7) + 5 ==> 1265

Divisibility test for 11 :

1 + 6 = 7

2 + 5 = 7

7 - 7 = 0

So, it is divisible by 11.

So, the answer is 1200 < N < 1300.

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