This problem is quite simple if you recall that the units place of powers of 2 follows a simple 4-step sequence. 

Observe the first few powers of 2:

2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64, 2⁷ = 128, 2⁸ = 256 . . .

The units place follows a sequence of 2, 4, 8, 6, 2, 4, 8, 6, etc. Thus, divide 102 by 4. This gives a remainder of 2.  

The second number in the sequence is 4, so the answer is 4.

For exponent problems like this, the easiest thing to do is to break down all the numbers that you have into their prime factors. Begin with the number given to you:

Now, in order for you to have a number that is a multiple of this, you will need to have at least  in the prime factorization of the given number.  For each of the answer choices, you have:

; This is the answer.

Because the numbers involved in your fraction are so large, you are going to need to do some careful manipulating to get your answer. (A basic calculator will not work for something like this.) These sorts of questions almost always work well when you isolate the large factors and notice patterns involved. Let's first focus on the numerator. Go ahead and break apart the  into its prime factors:

Note that these have a common factor of . Therefore, you can rewrite the numerator as:

Now, put this back into your fraction:

In Quantity A, we add the exponents because the operation is multiplication, so 4⁵ * 4^–3 = 4^5+(–3) = 4². In Quantity B, we subtract the exponents because the operation is division, so 4⁵ / 4^–3 = 4^5–(–3) = 4⁸. We don't have to finish multiplying out the exponents to see that Quantity B is greater.

This question is asking you for the ratio of m to n.  To figure it out, the easiest way is to figure out when 4 to an exponent equals 8 to an exponent.  The easiest way to do that is to list the first few results of 4 to an exponent and 8 to an exponent and check to see if any match up, before resorting to more drastic means of finding a formula.

And, would you look at that. .  Therefore, .

8 = 2³

(2³)^x = 2^3x

2^3x = 2^x+6  <- when the bases are the same, you can set the exponents equal to each other and solve for x

3x=x+6

2x=6

x=3

First rewrite the two expressions so that they have the same base, and then compare their exponents.

Combine exponents by multiplying: 

This is the same as the first given expression, so the two expressions are equal.

 can be written as  

Since there is a common base of , we can say

  or . 

The basees don't match.

However: 

 thus we can rewrite the expression as .

Anything raised to negative power means  over the base raised to the postive exponent. 

So, . . 

The bases don't match.

However: 

 and we recognize that .

Anything raised to negative power means  over the base raised to the postive exponent. 

.