### All GRE Math Resources

## Example Questions

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

Column A

5!/3!

Column B

6!/4!

**Possible Answers:**

The relationship cannot be determined from the information given.

The two quantities are equal.

The quantity in Column B is greater.

The quantity in Column A is greater.

**Correct answer:**

The quantity in Column B is greater.

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 #1 : Other Factors / Multiples

The prime factorization of 60 is?

**Possible Answers:**

2 * 2 * 3 * 5

2 * 3 * 5

2 * 3 * 10

2 * 3 * 5 * 2

**Correct answer:**

2 * 2 * 3 * 5

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?

**Possible Answers:**

**Correct answer:**3

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 #1 : How To Factor A Number

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

**Possible Answers:**

413(721 + 211)

211(413 + 721)

(721 + 211)(413 + 721)

721(413 + 211)

(721 + 413)(211 +721)

**Correct answer:**

721(413 + 211)

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 : Other Factors / Multiples

*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

**Possible Answers:**

The relationship cannot be determined from the information given.

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

**Correct answer:**

The relationship cannot be determined from the information given.

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

**Possible Answers:**

Quantity A is greater.

The relationship cannot be determined from the information given.

Quantity B is greater.

The two quantities are equal.

**Correct answer:**

Quantity A is greater.

32 = 2 * 16 = 2 * 4 * 4 = 2 * 2 * 2 * 2 * 2 = 2^{5}, so Quantity A = 5.

60 = 2 * 30 = 2 * 6 * 5 = 2 * 2 * 3 * 5 = 2^{2} * 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 ?

**Possible Answers:**

**Correct answer:**

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 #2 : How To Factor A Number

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

**Possible Answers:**

**Correct answer:**

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 #1 : How To Factor A Number

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

**Possible Answers:**

**Correct answer:**

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 2^{15}, which is the same as 4^{7} * 2; therefore, since you are asked for the largest integer value of *n*, 7 is your answer.

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

### Example Question #21 : Integers

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

**Possible Answers:**

**Correct answer:**

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: