If  is divisible by , then for some integer , . In each of the expressions, simplify, replace  for ,  then divide by  to determine whether the quotient is an integer:

Simplifying the expression:

Replacing  with :

Divide this by 17, and the result is

, an integer. 

Simplifying the expression:

Replacing  with :

Divide this by 17, and the result is

, an integer.

Simplifying the expression:

Replacing  with :

Divide this by 17, and the result is

, an integer.

Simplifying the expression:

Replacing  with :

Divide this by 17, and the result is

, an integer.

Simplifying the expression:

Replacing  with :

Divide this by 17, and the result is

, not an integer.

 is the correct choice.

The greatest integer that is less than or equal to  is .

The greatest integer that is less than or equal to  is .

The greatest integer that is less than or equal to  is .  is not correct as it is greater than.

Note that for positive decimal numbers, the floor function is the integer part of the number while for negative decimal numbers it is the integer part minus .

Firstly, we should start by noticing that  and  are co-prime, which means that they only share 1 as a common factor. It is due to the fact that they are consecutive integers. Also we should notice that  will never include  since it is the product from all integers from 1 to , therefore,  is not a factor of  

To find the greatest common factor of two numbers, we must first write their respective prime factorization, in other words, we must write these numbers as a product of prime numbers. This can be done by firstly dividing a given number by 2, then 3, and all the way up to higher prime numbers. 

 and 

Then we simply have to multiply all the common factors, with their lowest power:

The common factors here are only 5's, and the lowest power of 5 in both numbers is . Therefore, the greatest common factor is .

We start by writing the prime factorization of these two numbers:  and 

Note that to find the prime factorization there is not much to do other than just starting with the smallest prime numbers and the GMAT requires to know the prime numbers under 100.

The common factors are only 2's, and we apply to this common factor of the lowest power, which is 1. Therefore, the final answer is .

We start by writing the prime factorization of these two numbers: 

 and .

By taking the common factors and raising them to the lowest power, we get  or , which is the final answer.

For the least common multiple, we also need to write out the prime factorization of these numbers, as follows: 

 and .

Then, we multiply all the factors raised to their highest powers.

Here we have: , which is equal to .

We begin by writing the prime factorization of 78 and 36. 

 and .

The rule to find the least common multiple is to multiply all factors raised to their highest power.

Here we have:  or .

Let's start by writing the prime factorization of these two numbers: 

 and .

Then we just have to multiply each factor raised to its highet power for common factors, as follows:

 or .

For any positive integer  and ,  being their least common multiple and  being their greatest common factor, then .

Therefore,

  or .