## Example Questions

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### Example Question #1 : How To Factor A Number

Column A

5!/3!

Column B

6!/4!

The quantity in Column B is greater.

The two quantities are equal.

The quantity in Column A is greater.

The relationship cannot be determined from the information given.

The quantity in Column B is greater.

Explanation:

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.

### Example Question #11 : Factors / Multiples

The prime factorization of 60 is?

2 * 3 * 5

2 * 3 * 5 * 2

2 * 2 * 3 * 5

2 * 3 * 10

2 * 2 * 3 * 5

Explanation:

Prime numbers are numbers that can only divided by one and themselves.  Breaking 60 into its prime factors yields: ### Example Question #1 : Other Factors / Multiples

Which of the following integers are factors of both 24 and 42?

3
12
5
7
8
Explanation:

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.

### Example Question #4 : How To Factor A Number

721(413) + 211(721) is equal to which of the following?

413(721 + 211)

(721 + 413)(211 +721)

211(413 + 721)

(721 + 211)(413 + 721)

721(413 + 211)

721(413 + 211)

Explanation:

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

### Example Question #1 : How To Factor A Number

n is a positive integer .  p = 4 * 6 * 11 * n

Quantity A: The remainder when p is divided by 5

Quantity B: The remainder when p is divided by 33

Quantity B is greater.

Quantity A is greater.

The relationship cannot be determined from the information given.

The two quantities are equal.

The relationship cannot be determined from the information given.

Explanation:

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.

### Example Question #1 : How To Factor A Number

Quantitative Comparison

Quantity A: number of 2's in the prime factorization of 32

Quantity B: number of 2's in the prime factorization of 60

The relationship cannot be determined from the information given.

The two quantities are equal.

Quantity A is greater.

Quantity B is greater.

Quantity A is greater.

Explanation:

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.

### Example Question #1 : Other Factors / Multiples

If is an integer and is an integer, which of the following could be the value of ?      Explanation:

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 .

### Example Question #91 : Arithmetic

What is the sum of the individual factors of 100 and 200?      Explanation:

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. ### Example Question #9 : How To Factor A Number

What is the largest possible integer value of if divides 16! evenly?      Explanation:

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.

### Example Question #1 : How To Factor A Number

If the product of two distinct integer is , which of the following could not represent the sum of those two integers?      Explanation:

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:    ← Previous 1

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