Factors / Multiples - GRE Quantitative Reasoning

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 greatest common divisor of 64 and 14 is 2, as found by the prime factorization of 64 and 14. The least common multiple of 16 and 52 is 208, which can be found by looking at the decimal when 52 is divided by 16. The remainder is 0.25, or 1/4 so the fourth multiple of 52 is 208, which is also divisible by 16.

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".

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.

Factor each of our values into prime factors:

350 = 2 * 52 * 7

6270= 2 * 3 * 5 * 11 * 19

To find the least common multiple, we must choose the larger exponent for each of the prime factors involved. Therefore, we will select 2, 52, and 7 from 350 and 3, 11, and 19 from 6270.

Therefore, our least common multiple is 2 * 3 * 52 * 7 * 11 * 19 = 219,450.

Each of the numbers on the list must be able to "fit" (multiply evenly) into a larger number at the same time. I.e. the largest number (multiple) divided by any of the listed numbers will result in a whole number. For the coefficients, the maximum value is 10, and multiplying the highest two coefficients give us 60. Since 60 is divisible evenly by the lower values, we know that it is the least common multiple for the list. For the variables, both x and y will fit evenly into a theoretical number, "xy". We do not need an exponential version of this multiple as there are no exponents in the original list.

The least common multiple is the smallest number that is a multiple of all the numbers in the group. Let's list some multiples of the two numbers and find the smallest number in common to both.

multiples of 45: 45, 90, 135, 180, 225, 270, ...

multiples of 60: 60, 120, 180, 240, 300, 360, ...

The smallest number in common is 180.

First, find all of the factors of each number. Factors are the numbers that, like 16 and 24, can evenly be divided. The factors of 16 are 1, 2, 4, 8, 16. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.

Now, to find the greatest common factor, we find the largest number that is on both lists. This number is 8.

To make things easier, note 6930 is divisible by 30:

6930 = 231 * 30 = 3 * 77 * 3 * 2 * 5 = 3 * 7 * 11 * 3 * 2 * 5 = 2 * 32 * 5 * 7 * 11

288 = 2 * 144 = 2 * 12 * 12 = 2 * 2 * 2 * 3 * 2 * 2 * 3 = 25 * 32

Consider each of these "next to each other":

25 * 32

2 * 32 * 5 * 7 * 11

Each shares factors of 2 and 3. In the case of 2, they share 1 factor. In the case of 3, they share 2 factors. Therefore, their greatest common factor is: 2 * 32 = 2 * 9 = 18

The greatest common factor is the greatest factor that divides both numbers. To find the greatest common factor, first list the prime factors of each number.

18 = 2 * 3 * 3

24 = 2 * 2 * 2 * 3

18 and 24 share one 2 and one 3 in common. We multiply them to get the GCF, so 2 * 3 = 6 is the GCF of 18 and 24.

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:

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.

Let's go through the five statements.

The sum of two primes is always even: This is only true of the odd primes. 2 is also a prime number, however, and 2 plus an odd number is odd.

Every positive prime has a corresponding negative prime: This is also false. There are no negative primes. A prime number is defined as a number greater than 1 that is divisible by only 1 and itself.

There are only two primes that are consecutive positive integers on the number line: This is true and therefore the correct answer. 2 and 3 are the only primes that are consecutive. Because 2 is the only even prime, all other primes must have at least one number in between them (since every two odd numbers are separated by an even).

Multiplying two primes will always produce an odd number: This is also only true of odd primes. 2 * odd prime = even.

The distribution of primes is random: False. The primes are logarithmically distributed.

Quantity A: The smallest prime number is 2. We also need the least common multiple of 5 and 10, which is 10.

So Quantity A = 2 * 3 / 10 = 3/5

Quantity B: The smallest odd prime is 3. The second smallest odd prime is 5.

So Quantity B = 3 * 2 / 5 = 6/5

Quantity B is greater.

The prime factorization of 330 is .

The sum of the prime factors is .

23 is the only answer choice greater than 21.

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.