DIVISIBILITY RULES - FOR 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 AND PRIME DIVISORS 13 TO 47 AND ABOVE

Please study Multiplication Tables and
Division before Divisibility Rules
They are prerequisites here.

Here we study methods that can be used to determine
whether a number is evenly divisible by other numbers.

These are shortcuts for testing a number's factors
without resorting to division calculations.

The rules given below transform a given number's divisibilty
by a divisor to a smaller number's divisibilty by the same divisor.

If the result is not obvious after applying it once,
the rule should be applied again to the smaller number.

We present the rules with examples, in a simple way,
to follow, understand and apply.

We provide the rules for all the single digit divisors
and prime divisors upto 47 and above
and a general method which is simple to use.

Divisibility Rule for 2

A number is divisible by 2, if its last digit (units' digit) is divisible by 2.
i. e. if its last digit (units' digit) is any of the digits
0, 2, 4, 6 and 8 ( i. e. Even digits ).
i. e. All EVEN NUMBERS are divisible by 2.

Examples:

112 has last digit as 2 (even) and so is divisible by 2.
121 is not divisible by 2, since last digit is 1 (odd).

1234 is divisible by 2, since the last digit is 4 (even).
3003 is not divisible by 2, since last digit is 3 (odd).

95030 is divisible by 2, since the last digit is 0 (even).
68735 is not divisible by 2, since last digit is 5 (odd).

345617 is not divisible by 2, since last digit is 7 (odd).
231276 is divisible by 2, since the last digit is 6 (even).

1235678 is divisible by 2, since the last digit is 8 (even).
9814379 is not divisible by 2, since last digit is 9 (odd).

Divisibility Rule for 4

A number is divisible by 4, if the number formed by its last two digits
( i. e. tens' digit and units' digit) is divisible by 4.

Note : If the number is odd, there is no need to check the rule,
as odd number is not divisible by even number (4).

Example 1 of Divisibility Rule for 4

Find whether 4025 is divisible by 4 or not.

Solution :
As the number, 4025 is odd, it is not divisible by 4. Ans.

Example 2 of Divisibility Rule for 4

Find whether 234 is divisible by 4 or not.

Solution :
We verify whether the number formed by the last two digits (34)
is divisible by 4 or not.

              Dividend
 Divisor   4 )   34   ( 8   Quotient    
                 32
                 ---
                  2   Remainder 
                 ---  

Example 3 of Divisibility Rule for 4

Find whether 5678 is divisible by 4 or not.

Solution :
We verify whether the number formed by the last two digits (78)
is divisible by 4 or not.

              Dividend
 Divisor   4 )   78   ( 19   Quotient    
                 4
                 ---
                 38
                 36    
                 ---  
                  2   Remainder 
                 ---   

Example 4 of Divisibility Rule for 4

Find whether 81756 is divisible by 4 or not.

Solution :
We verify whether the number formed by the last two digits (56)
is divisible by 4 or not.

              Dividend
 Divisor   4 )   56   ( 14   Quotient    
                 4
                 ---
                 16
                 16    
                 ---  
                  0   Remainder 
                 ---   

Example 5 of Divisibility Rule for 4

Find whether 987432 is divisible by 4 or not.

Solution :
We verify whether the number formed by the last two digits (32)
is divisible by 4 or not.

              Dividend
 Divisor   4 )   32   ( 8   Quotient    
                 32
                 ---
                  0   Remainder 
                 ---  

Thus we can check divisible by 4 of any number,
how big it may be, by simply checking the
divisibilty of the number formed by the last two digits.

Divisibility Rule for 8

A number is divisible by 8, if the number formed by its last three digits
(hundreds' digit, tens' digit and units' digit) is divisible by 8.

Note : If the number is odd, there is no need to check the rule,
as odd number is not divisible by even number (8).

Example 1 of Divisibility Rule for 8

Find whether 45567 is divisible by 8 or not.

Solution :
As the number, 45567 is odd, it is not divisible by 8. Ans.

Example 2 of Divisibility Rule for 8

Find whether 47384 is divisible by 8 or not.

Solution :
We verify whether the number formed by the last three digits (384)
is divisible by 8 or not.

              Dividend
 Divisor   8 )   384   ( 48   Quotient    
                 32
                 ---
                  64
                  64    
                 ---  
                   0   Remainder 
                 ---   

Example 3 of Divisibility Rule for 8

Find whether 289512 is divisible by 8 or not.

Solution :
We verify whether the number formed by the last three digits (512)
is divisible by 8 or not.

              Dividend
 Divisor   8 )   512   ( 64   Quotient    
                 48
                 ---
                  32
                  32    
                 ---  
                   0   Remainder 
                 ---   

Example 4 of Divisibility Rule for 8

Find whether 1756942 is divisible by 8 or not.

Solution :
We verify whether the number formed by the last three digits (942)
is divisible by 8 or not.

              Dividend
 Divisor   8 )   942   ( 117   Quotient    
                 8
                 ---
                 14
                  8    
                ----  
                  62    
                  56
                ----
                   6    Remainder
                ----

Example 5 of Divisibility Rule for 8

Find whether 19230404 is divisible by 8 or not.

Solution :
We verify whether the number formed by the last three digits (404)
is divisible by 8 or not.

              Dividend
 Divisor   8 )   404   ( 50   Quotient    
                 40
                 ---
                  04
                   0    
                 ---  
                   4   Remainder 
                 ---   

Thus we can check divisible by 8 of any number,
how big it may be, by simply checking the
divisibilty of the number formed by the last three digits.

Divisibility Rule for 5

A number is divisible by 5, if its last digit (units' digit) is either 0 or 5.

Examples:

5342 is not divisible by 5, since last digit is neither 0 nor 5.
4235 is divisible by 5, since last digit is 5.

2345 is divisible by 5, since last digit is 5.
7530 is divisible by 5, since last digit is 0.

3075 is divisible by 5, since last digit is 5.
3057 is not divisible by 5, since last digit is neither 0 nor 5.

5120 is divisible by 5, since last digit is 0.
5012 is not divisible by 5, since last digit is neither 0 nor 5.

98760 is divisible by 5, since last digit is 0.
67453 is not divisible by 5, since last digit is neither 0 nor 5.

Divisibility Rule for 10

A number is divisible by 10, if its last digit (units' digit) is 0.

Examples:

The numbers 70, 900, 5680, , 20000, 37610
are divisible by 10 as all have the last digit 0.

The numbers 97, 121, 5008, 98761, 605009
are not divisible by 10, as they do not have 0 in the units' place.

Divisibility Rule for 3 :

A number is divisible by 3, if the number obtained by adding its digits is divisible by 3.
If you don't know the new number's divisibility, you can apply the rule again.

Find whether 23451 is divisible by 3 or not.

Solution :
The number formed by sum of the digits = 2 + 3 + 4 + 5 + 1 = 15.
We know 15 is divisible by 3.
∴ 23451 is divisible by 3. Ans.

Example 2 of Divisibility Rule for 3

Find whether 142367 is divisible by 3 or not.

Solution :
The number formed by sum of the digits = 1 + 4 + 2 + 3 + 6 + 7 = 23.
We know 23 is not divisible by 3.
∴ 142367 is not divisible by 3. Ans.

Example 3 of Divisibility Rule for 3

Find whether 652743 is divisible by 3 or not.

Solution :
The number formed by sum of the digits = 6 + 5 + 2 + 7 + 4 + 3 = 27.
We know 27 is divisible by 3.
∴ 652743 is divisible by 3. Ans.

Example 4 of Divisibility Rule for 3

Find whether 5230716 is divisible by 3 or not.

Solution :
The number formed by sum of the digits = 5 + 2 + 3 + 0 + 7 + 1 + 6 = 24.
We know 24 is divisible by 3.
∴ 5230716 is divisible by 3. Ans.

Example 5 of Divisibility Rule for 3

Find whether 91451080 is divisible by 3 or not.

Solution :
The number formed by sum of the digits = 9 + 1 + 4 + 5 + 1 + 0 + 8 + 0 = 29.
We know 28 is not divisible by 3.
∴ 91451080 is not divisible by 3. Ans.

Divisibility Rule for 6

A number is divisible by 6, if the number is divisible by both 2 and 3.

Examples:
check whether the Numbers in examples under divisibility by 3, are divisible by 6 or not.

Solution:
Here, we have to first check whether the number is even or odd.
If it is odd, then it is not divisible by 6.

If it is even, then check for the divisibility by 3. If it is divisible by 3,
then it is divisible by 6. Otherwise not.

In the above examples under divisibility by 3,
Numbers in Examples 1, 2, 3 are odd numbers
and hence are not divisible by 6.

Out of Numbers in remaining Examples 4 and 5,
Number in Example 5 is not divisible by 3.

∴ only Number in Example 4 is divisible by 6.

Divisibility Rule for 9

A number is divisible by 9, if the number obtained by adding its digits is divisible by 9.
If you don't know the new number's divisibility, you can apply the rule again. (usually, it is not required to apply again.)

Example 1 of Divisibility Rule for 9

Find whether 23454 is divisible by 9 or not.

Solution :
The number formed by sum of the digits = 2 + 3 + 4 + 5 + 4 = 18.
We know 18 is divisible by 9.
∴ 23454 is divisible by 9. Ans.

Example 2 of Divisibility Rule for 9

Find whether 641857 is divisible by 9 or not.

Solution :
The number formed by sum of the digits = 6 + 4 + 1 + 8 + 5 + 7 = 31.
We know 31 is not divisible by 9.
∴ 641857 is not divisible by 9. Ans.

Example 3 of Divisibility Rule for 9

Find whether 652743 is divisible by 9 or not.

Solution :
The number formed by sum of the digits = 6 + 5 + 2 + 7 + 4 + 3 = 27.
We know 27 is divisible by 9.
∴ 652743 is divisible by 9. Ans.

Example 4 of Divisibility Rule for 9

Find whether 5230716 is divisible by 9 or not.

Solution :
The number formed by sum of the digits = 5 + 2 + 3 + 0 + 7 + 1 + 6 = 24.
We know 24 is not divisible by 9.
∴ 5230716 is not divisible by 9. Ans.

Example 5 of Divisibility Rule for 9

Find whether 91451087 is divisible by 9 or not.

Solution :
The number formed by sum of the digits = 9 + 1 + 4 + 5 + 1 + 0 + 8 + 7 = 36.
We know 36 is divisible by 9.
∴ 91451087 is divisible by 9. Ans.

Divisibility Rule for 7

To find out, if a number is divisible by 7, take the last digit, double it, and subtract it from the rest of the number. If you get an answer divisible by 7 (including zero), then the original number is divisible by 7.
If you don't know the new number's divisibility, you can apply the rule again.

Example 1 of Divisibility Rule for 7

Find whether 343 is divisible by 7 or not.

Solution :
Twice the last digit = 2 x 3 = 6; Rest of the number = 34
Subtracting, 34 - 6 = 28 is divisible by 7. ( 28 = 4 x 7)
∴ 343 is divisible by 7. Ans.

Example 2 of Divisibility Rule for 7

Find whether 8965 is divisible by 7 or not.

Solution :
Twice the last digit = 2 x 5 = 10; Rest of the number = 896
Subtracting, 896 - 10 = 886
To check whether 886 is divisible by 7 :
Twice the last digit = 2 x 6 = 12; Rest of the number = 88
Subtracting, 88 - 12 = 66 is not divisible by 7. ( 66 = 9 x 7 + 3)
∴ 8965 is not divisible by 7. Ans.

Example 3 of Divisibility Rule for 7

Find whether 49875 is divisible by 7 or not.

Solution :
To check whether 49875 is divisible by 7 :
Twice the last digit = 2 x 5 = 10; Rest of the number = 4987
Subtracting, 4987 - 10 = 4977 To check whether 4977 is divisible by 7 :
Twice the last digit = 2 x 7 = 14; Rest of the number = 497
Subtracting, 497 - 14 = 483

To check whether 483 is divisible by 7 :
Twice the last digit = 2 x 3 = 6; Rest of the number = 48
Subtracting, 48 - 6 = 42 is divisible by 7. ( 42 = 6 x 7 )

∴ 49875 is divisible by 7. Ans.

Example 4 of Divisibility Rule for 7

Find whether 987651 is divisible by 7 or not.

Solution :
To check whether 987651 is divisible by 7 :
Twice the last digit = 2 x 1 = 2; Rest of the number = 98765
Subtracting, 98765 - 2 = 98763 To check whether 98763 is divisible by 7 :
Twice the last digit = 2 x 3 = 6; Rest of the number = 9876
Subtracting, 9876 - 6 = 9870

To check whether 987 is divisible by 7 :
Twice the last digit = 2 x 7 = 14; Rest of the number = 98
Subtracting, 98 - 14 = 84

To check whether 84 is divisible by 7 :
Twice the last digit = 2 x 4 = 8; Rest of the number = 8
Subtracting, 8 - 8 = 0 is divisible by 7.

∴ 987651 is divisible by 7. Ans.

Example 5 of Divisibility Rule for 7

Find whether 986953 is divisible by 7 or not.

Solution :
To check whether 986953 is divisible by 7 :
Twice the last digit = 2 x 3 = 6; Rest of the number = 98695
Subtracting, 98695 - 6= 98689 To check whether 98689 is divisible by 7 :
Twice the last digit = 2 x 9 = 18; Rest of the number = 9868
Subtracting, 9868 - 18 = 9850

To check whether 985 is divisible by 7 : Twice the last digit = 2 x 5 = 10; Rest of the number = 98
Subtracting, 98 - 10 = 88

To check whether 88 is divisible by 7 : Twice the last digit = 2 x 8 = 16; Rest of the number = 8
Subtracting, 8 - 16 = -8 is not divisible by 7.

∴ 986953 is not divisible by 7. Ans.

Divisibility Rule for 11

To find out, if a number is divisible by 11, take the sum of the digits at odd places, and the sum of the digits at even places, and subtract one from the other. If you get an answer divisible by 11 (including zero), then the original number is divisible by 11.
If you don't know the new number's divisibility, you can apply the rule again. (usually, it is not required to apply again.)

Example 1 of Divisibility Rule for 11

Find whether 65714 is divisible by 11 or not.

Solution :
sum of alternate digits = 6 + 7 + 4 = 17; sum of remaining digits = 5 + 1 = 6;
Subtracting, 17 - 6 = 11 is divisible by 11.
∴ 65714 is divisible by 11. Ans.

Example 2 of Divisibility Rule for 11

Find whether 123453 is divisible by 11 or not.

Solution :
sum of alternate digits = 1 + 3 + 5 = 9; sum of remaining digits = 2 + 4 + 3 = 9;
Subtracting, 9 - 9 = 0.
∴ 123453 is divisible by 11. Ans.

Example 3 of Divisibility Rule for 11

Find whether 5232819 is divisible by 11 or not.

Solution :
sum of alternate digits = 5 + 3 + 8 + 9 = 25; sum of remaining digits = 2 + 2 + 1 = 5;
Subtracting, 25 - 5 = 20 is not divisible by 11.
∴ 5232819 is not divisible by 11. Ans.

Example 4 of Divisibility Rule for 11

Find whether 58976544 is divisible by 11 or not.

Solution :
sum of alternate digits = 5 + 9 + 6 + 4 = 24; sum of remaining digits = 8 + 7 + 5 + 4 = 24;
Subtracting, 24 - 24 = 0 is divisible by 11.
∴ 58976544 is divisible by 11. Ans.

Example 5 of Divisibility Rule for 11

Find whether 914510876 is divisible by 11 or not.

Solution :
sum of alternate digits = 9 + 4 + 1 + 8 + 6 = 28; sum of remaining digits = 1 + 5 + 0 + 7 = 13;
Subtracting, 28 - 13 = 15 is not divisible by 11.
∴ 914510876 is not divisible by 11. Ans.

Exercise on Divisibility Rules for 2 to 11

(1) Check whether the following numbers are divisible by 2.
(i) 97234 (ii) 16524 (iii) 3825 (iv) 52618 (v) 843071
(vi) 64127 (vii) 123612 (viii) 46782 (ix) 2613 (x) 273842

(2) Check whether the following numbers are divisible by 8.
(i) 1234896 (ii) 23468 (iii) 56818 (iv) 789248 (v) 987364
(vi) 34168 (vii) 987167 (viii)123856 (ix) 45060840 (x) 123678

(3) Check whether the numbers in Question (2) are divisible by 4.
(i) 1234896 (ii) 23468 (iii) 56818 (iv) 789248 (v) 987364
(vi) 34168 (vii) 987167 (viii)123856 (ix) 45060840 (x) 123678

(4) Check whether the following numbers are divisible by 3.
(i)3624 (ii)94506 (iii)97531(iv)105756 (v)739285
(vi)12369 (vii)456786 (viii)964152 (ix)1234 (x)24566

(5) Check whether the following numbers are divisible by 9.
(i)81756 (ii)77782 (iii)672588 (iv)5678 (v)517248
(vi)97281 (vii)45638 (viii)909082 (ix)12339 (x)64826

(6) Check whether the following numbers are divisible by 6.
(i)753216 (ii)453212 (iii)11111112 (iv)28524 (v)94506
(vi)65274 (vii)832451 (viii)256784 (ix)28584 (x)142367

(7) Check whether the following numbers are divisible by 5.
(i) 3465 (ii) 9870 (iii) 99184 (iv) 76513 (v) 8710
(vi) 73969 (vii) 34785 (viii) 45780 (ix) 50000 (x) 10234

(8) Check whether the numbers in Question (7) are divisible by 10.
(i) 3465 (ii) 9870 (iii) 99184 (iv) 76513 (v) 8710
(vi) 73969 (vii) 34785 (viii) 45780 (ix) 50000 (x) 10234

(9) Check whether the following numbers are divisible by 7.
(i)994 (ii)8267 (iii)2799 (iv)5508 (v)1239
(vi)49874 (vii)78498 (viii)2793 (ix)123031 (x) 5432

(10) Check whether the following numbers are divisible by 11.
(i)36553 (ii)987654 (iii)352715 (iv)789564 (v)43923
(vi)247963 (vii)391783 (viii)25344 (ix)571394 (x)13574

For Answers see at the bottom of the page.

The following discussion requires the knowledge of Basic Algebra

Divisibility Rule for any prime number 'p'

To find out, if a number is divisible by 'p', take the last digit, multiply it by 'n', and add it to the rest of the number.
or multiply it by '(p - n)' and subtract it from the rest of the number.
If you get an answer divisible by 'p' (including zero), then the original number is divisible by 'p'.
If you don't know the new number's divisibility, you can apply the rule again.

Finding 'n' for different prime numbers :

'n' is such that 'p' is the least factor of (10n - 1)
or n = (kp + 1)/10 where k is the least natural number for which
(kp + 1)/10 is a natural number (= n) or (kp + 1) is a multiple of 10.

For any prime number 'p', check for what value of
k( = 1, 2, 3, 4, .......), (kp + 1) is a multiple of 10.

For Example

For 'p' = 13, (3)13 + 1 = 40 = a multiple of 10. So k = 3.
∴ n = (kp + 1)⁄10 = 40⁄10 = 4

For 'p' = 19, (1)19 + 1 = 20 = a multiple of 10. So k = 1.
∴ n = (kp + 1)⁄10 = 20⁄10 = 2

For 'p' = 29, (1)29 + 1 = 30 = a multiple of 10. So k = 1.
∴ n = (kp + 1)⁄10 = 30⁄10 = 3

The values of 'n' for prime numbers below 50 are

For 13, n = 4 and (p - n) = 13 - 4 = 9
For 17, n = 12 and (p - n) = 17 - 12 = 5
For 19, n = 2 and (p - n) = 19 - 2 = 17
For 23, n = 7 and (p - n) = 23 - 7 = 16
For 29, n = 3 and (p - n) = 29 - 3 = 26

For 31, n = 28 and (p - n) = 31 - 28 = 3
For 37, n = 26 and (p - n) = 37 - 26 = 11
For 41, n = 37 and (p - n) = 41 - 37 = 4
For 43, n = 13 and (p - n) = 43 - 13 = 30
For 47, n = 33 and (p - n) = 47 - 33 = 14

Note that for multiplying the last digit,
for some prime numbers 'n' is more convenient and
for some others '(p - n)' is more convenient.

If you select 'n', add ('n' x last digit), to the rest of the number.
If you select '(p - n)', subtract {(p - n) x last digit},
from the rest of the number.

For example, for 19, n = 2 is more covenient than (p - n) = 17.

So, the divisibilty rule for 19 is as follows.

To find out, whether a number is divisible by 19, take the last digit, multiply it by 2, and add it to the rest of the number. If you get an answer divisible by 19 (including zero), then the original number is divisible by 19.
If you don't know the new number's divisibility, you can apply the rule again.

For 31, (p - n) = 3 is more covenient than n = 28.

So, the divisibilty rule for 31 is as follows.

To find out, whether a number is divisible by 31, take the last digit, multiply it by 3, and subtract it from the rest of the number. If you get an answer divisible by 31 (including zero), then the original number is divisible by 31.
If you don't know the new number's divisibility, you can apply the rule again.

Look at the divisibility rule for 7 above.
There, we chose, '(p - n)' = 2 for 7.
We can also choose 'n = 5' for 7 in which case
we have to multiply the last digit by 5 and add it
to the rest of the number to check its divisibilty by 7.

The method of finding 'n' given above
can be extended to prime numbers above 50 also.

General divisibility rule

Let 'd' ( > 10 ) be the divisor and 'D' ( > 10 ) be the dividend.
We want to check the divisibilty of D by d.

Let D₁ be the units' digit
and D₂ be the rest of the number of D.
i.e. D = D₁ + 10D₂

Let d₁ be the units' digit
and d₂ be the rest of the number of d.
i.e. d = d₁ + 10d₂

To find out, whether D is divisible by d, find (D₂d₁ - D₁d₂) If you get an answer divisible by d (including zero), then D is divisible by d.
If you don't know the new number's divisibility, you can apply the rule again.

Write D₂, D₁, d₂, d₁ as follows.

D₂     D₁

d₂     d₁

Cross multiply and take the difference.
If this difference is divisible by d, then D is divisible by d.

If you don't know the new number's divisibility,
you can apply the rule again.

Let us see an example.

Find whether 8827 is divisible by 91.

Solution :

882     7

  9     1

(882)(1) - (7)(9) = 882 - 63 = 819

81     9

9     1

(81)(1) - (9)(9) = 81 - 81 = 0

∴ 8827 is divisible by 91. Ans.

Proof of Divisibilty Rule for 7

Let 'D' ( > 10 ) be the dividend.

Let D₁ be the units' digit
and D₂ be the rest of the number of D.
i.e. D = D₁ + 10D₂

We have to prove
(i) if D₂ - 2D₁ is divisible by 7,
then D is also divisible by 7

and (ii) if D is divisible by 7,
then D₂ - 2D₁ is also divisible by 7.

Proof of (i) :
D₂ - 2D₁ is divisible by 7 ⇒ D₂ - 2D₁ = 7k where k is any natural number.
Multiplying both sides by 10, we get
10D₂ - 20D₁ = 70k
Adding D₁ to both sides, we get
(10D₂ + D₁) - 20D₁ = 70k + D₁
⇒ (10D₂ + D₁) = 70k + D₁ + 20D₁
⇒ D = 70k + 21D₁ = 7(10k + 3D₁) = a multiple of 7.
⇒ D is divisible by 7. (proved.)

Proof of (ii) :
D is divisible by 7 ⇒ D₁ + 10D₂ is divisible by 7 ⇒ D₁ + 10D₂ = 7k where k is any natural number.
Subtracting 21D₁ from both sides, we get
10D₂ - 20D₁ = 7k - 21D₁
⇒ 10(D₂ - 2D₁) = 7(k - 3D₁)
⇒ 10(D₂ - 2D₁) is divisible by 7
Since 10 is not divisible by 7,
(D₂ - 2D₁) is divisible by 7. (proved.)

In a similar fashion,
we can prove the divisibilty rule for any prime divisor.

Answers to Exercise on Divisibility Rules for 2 to 11 :

(1)(i) yes (ii) yes (iii) no (iv) yes (v) no
(vi) no (vii) yes (viii) yes (ix) no (x) yes

(2) (i) yes(ii) no(iii) no(iv) yes(v) no
(vi) yes (vii) no (viii) yes (ix) yes (x) no

(3) (i) yes (ii) yes (iii) no (iv) yes (v) yes
(vi) yes (vii) no (viii) yes (ix) yes (x) no

(4) (i) yes (ii) yes (iii) no (iv) yes (v) no
(vi) yes (vii) no (viii) yes (ix) no (x) no

(5) (i) yes (ii) no (iii) yes (iv) no (v) yes
(vi) yes (vii) no (viii) no (ix) yes (x) no

(6) (i) yes (ii) no (iii) yes (iv) yes (v) no
(vi) yes (vii) no (viii) no (ix) yes (x) no

(7) (i) yes (ii) yes (iii) no (iv) no (v) yes
(vi) no (vii) yes (viii) yes (ix) yes (x) no

(8) (i) no (ii) yes (iii) no (iv) no (v) yes
(vi) no (vii) no (viii) yes (ix) yes (x) no

(9) (i) yes (ii) yes (iii) no (iv) no (v) yes
(vi) no (vii) yes (viii) yes (ix) no (x) yes

(10) (i) yes (ii) no (iii) yes (iv) no (v) yes
(vi) no (vii) no (viii) yes (ix) no (x) yes