Factors / Multiples - GRE Quantitative Reasoning

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:

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

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.

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.

This question tests basic number properties. Prime numbers are numbers which are divisible only by one and themselves. Answer options '2' and '4' are automatically out, because they will always produce even products with a and b, and the sum of two even products is always even. Since no even number greater than 2 is prime, 2 and 4 cannot be answer options. 3 is tempting, until you remember that the sum of any two multiples of 3 is itself divisible by 3, thereby negating any possible answer for c except 3, which is impossible. There are, however, several possible combinations that work with x = 1. For instance, a = 8 and b = 9 means that 8(1) + 9(1) = 17, which is prime. You only need to find one example to demonstrate that an option works. This eliminates the "None of the other answers" option as well.

The third smallest prime number is 5. (Don't forget that 2 is a prime number, but 1 is not!)

The smallest two digit prime number is 11.

Now we can evaluate the entire expression:

a prime number is divisible by itself and 1 only

list the factors of each number:

6: 1,2,3,6

9: 1,3,9

71: 1,71

51: 1, 3,17,51

15: 1,3,5,15

The definition of a prime number is a number that is divisible by only one and itself. A prime number can't be divided by zero, because numbers divided by zero are undefined. The smallest prime number is 2, which is also the only even prime.

Pick a prime number to see that 3_x_ is not always even, for example 3 * 3 = 9.

But 2 is a prime number as well, so 3 * 2 = 6 which is even, so we can't say that 3_x_ is either even or odd.

Neither 9 nor 6 in our above example is prime, so 3_x_ is not a prime number.

Lastly, 9 is not divisible by 4, so 3_x_ is not always divisible by 4.

Therefore the answer is "Cannot be determined".

For starters, 1 is not a prime number, so eliminate the answer choices with 1 in them. Even if you have no idea what twin primes are, at least you've narrowed down the possibilities.

Twin primes are consecutive prime numbers with one even number in between them. 3 and 5 is the only set of twin primes listed. 2 and 3 are not separated by any numbers, and 13 and 19 are not consecutive primes, nor are they separated by one even number only. You should do your best to remember definitions and formulas such as this one, because these questions are considered "free" points on the test. There is no real math involved, just something to remember! Being able to answer a question like this quickly will give you more time for the computationally advanced problems.

As we go up on the number line, the number of primes decreases almost exponentially. Therefore there are far more prime numbers between 0 and 100 than there are between 101 and 200. This is a general number theory point that is important to know, but trying to come up with some primes in these two groups will also quickly demonstrate this principle.

All of these statements are true. Let's go through them.

1. There are no negative prime numbers. This appears as if it might be false, but in fact, the prime numbers are defined as whole numbers greater than one that are divisible by only one and itself.

2. Every number except 0 and 1 is a prime number or product of primes. This is also true. Let's look at the factorization of a number that isn't prime. For example, 6 = 2 * 3, which is a product of primes. 12 = 2 * 2 * 3, which is also a product of primes.

3. Every number has a unique prime factorization. We just saw that every number is either prime, or a product of primes. Therefore each number must have a unique prime factorization. Just as above, 6 is the product of two primes, 2 and 3. No other number can be made by mulitplying 2 * 3. The same is true for 12. When we multiply 2 * 2 * 3, the only number we will ever get is 12.