Solution:

21982030

Explanation:
Next number formed is (number * (number-2)) - number;
i.e
(5 * 3) - 5 => 10
(10 * 8) - 10 => 70
(70 * 68) - 70 => 4690
(4690 * 4688) - 4690 => 21982030

Solution:

5 => FIVE

Half of five => Remove 2 letters out of 4 i.e Remove letter 'F' AND 'E'.
It remains, 'IV'
The Roman value of 'IV' is 4.

Solution:

13 numbers

Explanation:
Since we are looking to find odd reminder which implies that the remainder will be either 1 or 3
Therefore number must follow below two conditions:
A) We can list number 5n + 1 or 5n + 3. (where n = 0,1,2,3.....)
B) The number must end in 1 or 3.

List of such numbers less than 100 are:
1, 3, 11, 13, 21, 23, 31, 33, 41, 43, 51, 53, 61, 63, 71, 73, 81, 83, 91, 93

Let's remove the nonprime ones which are 1, 21, 33, 51, 63, 81 and 91

So, I will be left with 13 numbers which fit the bill.
3, 11, 13, 23, 31, 41, 43, 53, 61, 71, 73, 83 and 93

Solution:

There can be many solutions. Three such solutions are:
A) ((8 - 8)^8)!
B) sqrt(8) * sqrt(8)/8
C) (8*(8 - 8))!

Solution:

The equation can be solved by substituting
B = 7
A = 8
S = 8
E = 3
L = 5

Therefore the equation now looks as:
7483 +
7455
-------
14938
-------

Solution:

A = B = K =6
C =7
D = H = 1,
E=0 ,
F = I = G = 4,
J=3,
L=5

Now we can rewrite the equations as :
66 = 76 - 10
44 = 60 - 14
16 = 64 - 46
37 = 67 - 30
35 = 53 - 16
10 = 74 - 64

Solution:

NUMBER +
NUMBER
=======
PUZZLE

Above puzzle can be solved by substituting
N - 2
U - 0
M - 1
B - 6
E - 8
R - 9
P - 4
Z - 3
L - 7

The puzzle can now be read as:

201689 +
201689
----------
403378

Solution:

45-6 = 78 / 2 = 39 * 1

Explanation:
A = 4
B = 5
C = 6
D = 7
E = 8
F = 2
G = 3
H = 9
I = 1

So below equation can be solved as:
AB - C = DE / F = GH * I
45 - 6 = 78/2 = 39 * 1
39 = 39 = 39.

Solution:

There can be two possible solutions as shown in the picture below.

1,19,3,12,16,14,18,10,7,6,15,2,5,9,13,17,8,11,4
1,19,3,18,16,14,12,4,13,6,9,8,17,15,7,5,2,11,10

Solution:

As you see in the image, we had to move just the circle number 1, 7 and 10 and it is done.