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.

,

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

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

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

Now we can add our two sums.

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.

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.

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.

Prime numbers are numbers that can only divided by one and themselves. Breaking 60 into its prime factors yields:

Because , the answer choice that has a factorization set that cancels out completely with 396 will produce an integer. Only 18 fits this qualification, since .

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.

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 :

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

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 could be factored:

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

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 or is an acceptable answers, because both the integers would be equivalent and not distinct.

Since is negative, it is the product of a positive and negative integer. Consider all of the ways that it could be factored, and the sums these factors would produce:

is the answer choice that matches.