This is a basic factorial question. A factorial is equal to the number times every positive whole number smaller than itself. In Column A, the numerator is 5 * 4 * 3 * 2 * 1 while the denominator is 3 * 2 * 1.

As you can see, the 3 * 2 * 1 can be cancelled out from both the numerator and denominator, leaving only 5 * 4.

The value for Column A is 5 * 4 = 20.

In Column B, the numerator is 6 * 5 * 4 * 3 * 2 * 1 while the denominator is 4 * 3 * 2 * 1. After simplifying, Column B gives a value of 6 * 5, or 30.

Thus, Column B is greater than Column A.

3 is the only answer that is a factor of both 24 and 42. 42/3 = 14 and 24/3 = 8.  The other answers are either a factor of 24 OR 42 or neither, but not both.

The answer is 721(413 + 211) because we can pull out a common factor, or 721, from both sides of the equation.

Let's consider Quantity B first.  What will the remainder be when p is divided by 33?

4, 6 and 11 are factors of p which means that 2 * 2 * 2 * 3 * 11 * n will equal p.  We can group the 3 and 11 to see that 33 will always be a factor of p and will have no remainder. Thus Quantity B will always equal 0 no matter the value of n.  

Now consider Quantity A. Let's consider first the values for p when n equals 1 through 5. When n = 1, p = 264, and the remainder is 4/5 or 0.8. 

n = 2, p = 528, and the remainder is 3/5 or 0.6.

n = 3, p = 792, and the remainder is 2/5 or 0.4.

n = 4, p = 1056, and the remainder is 1/5 or 0.2.

n = 5, p = 1320, and the remainder is 0 (because when n = 5, 5 becomes a factor of p and thus there is no remainder.

Because Quantity A can be equal to or greater than B, there is not enough information given to determine the relationship.

32 = 2 * 16 = 2 * 4 * 4 = 2 * 2 * 2 * 2 * 2 = 2⁵, so Quantity A = 5.

60 = 2 * 30 = 2 * 6 * 5 = 2 * 2 * 3 * 5 = 2² * 3 * 5, so Quantity B = 2.

Quantity A is greater.  Even though 60 is a larger number than 32, 32 has more 2's in its prime factorization.

Because $\displaystyle \small 396=2*2*3*3*11$, the answer choice that has a factorization set that cancels out completely with 396 will produce an integer. Only 18 fits this qualification, since $\displaystyle \small 18=2*3*3$.

Do not try to count out the factors. A neat formula for finding the sum of factors of a number can be utilized by first determining the prime factorization of the number.

$\displaystyle s = \frac{(a^{x+1} - 1)(b^{y+1} - 1)(c^{z+1} - 1)}{(a - 1)(b - 1)(c - 1)}$, 

where s is the sum, a, b, and c are factors, and x, y, and z are the powers of these factors.

$\displaystyle 100=4*25=2^2*5^2$

Then, a = 2, b = 5, x = 2, y = 2.

$\displaystyle s_{100}=\frac{(2^{2+1}-1)(5^{2+1}-1)}{(2-1)(5-1)}$

$\displaystyle s_{100}=\frac{(7)(124)}{(1)(4)}=\frac{868}{4}=217$

 $\displaystyle 200=8*25=2^3*5^2$

Then, a = 2, b = 5, x = 3, y = 2.

$\displaystyle s_{200}=\frac{(2^{3+1}-1)(5^{2+1}-1)}{(2-1)(5-1)}$

$\displaystyle s_{200}=\frac{(15)(124)}{(1)(4)}=\frac{1860}{4}=465$

Now we can add our two sums.

$\displaystyle s_{100}+s_{200}=217+465=682$

This question is really asking, “How many factors of 4 are there in 16!”?  To ascertain this, list all the even numbers and count the total number of 2s among those factors.

Respectively, 16, 14, 12, 10, 8, 6, 4, 2 have 4, 1, 2, 1, 3, 1, 2, 1 factors of 2. 

The total then is 15. This means that you have a factor of 2¹⁵, which is the same as 4⁷ * 2; therefore, since you are asked for the largest integer value of n, 7 is your answer.

Any larger integer value would not allow 4ⁿ to divide 16! evenly.

Since we're dealing with a product that comes out to a positive value, it could be the product of two positives or two negatives.

That being said, consider the ways we could factor $\displaystyle 143$:

$\displaystyle 1\cdot143$

$\displaystyle -1\cdot-143$

$\displaystyle 11\cdot 13$

$\displaystyle -11\cdot -13$

For each of these four possible factors, there are four possible sums:

$\displaystyle 144$

$\displaystyle -144$

$\displaystyle 24$

$\displaystyle -24$

When the product of two numbers is positive, that means that either both numbers were positive, or both numbers were negative.

Now, considering the way $\displaystyle 36$ could be factored:

$\displaystyle 1 * 36$

$\displaystyle 2*18$

$\displaystyle 3*12$

$\displaystyle 4*9$

$\displaystyle 6*6$

And of course the cases where both values are negative. For each of these potential factors, the sums are then

$\displaystyle |1+36|=|37|$

$\displaystyle |2+18|=|20|$

$\displaystyle |3+12|=|15|$

$\displaystyle |4+9|=|13|$

$\displaystyle |6+6|=|12|$

Absolute value signs are used to denote that either a sum or it's negative suffices. However, recall that we're told the two integers are distinct!

Due to this, neither $\displaystyle -12$ or $\displaystyle 12$ is an acceptable answers, because both the integers would be equivalent and not distinct.