Three is the largest number that can be divided evenly into 27 and the positive integer x, while 10 is the largest number that can be divided evenly into both 100 and x. Which of the following is the largest possible number that could be divided into x and 2500  Choose the correct option.
A) 30
B) 50
C) 210
D) 300
E) 700
27 = 3*3*3, and its GCF with x is 3, which implies that x has ONE factor of 3, but not TWO factors of 3.   100 and x have a GCF of 10, which implies that x has ONE factor of 10, but not TWO factors of 10.   Then we want to know what is the largest possible GCF of x and 2100.   Well 2500 = 5 * 5 * 10 * 10   We want x to include as many factors in common with 2100 as possible, to make the GCF with 2100 as big as possible.   We know x has one factor of 3, but not two factors --- that takes the 3.   We know x has one factor of 10, but not two factors --- we can take one of those 10's, but we have to leave the other   No other restrictions, so we can also grab that factor of 7 --- nothing saying that we can't, and it's there for the taking.   5*10= 
<gadget id="calculator">5*10</gadget>
<output>50</output>
50   If we allow x to include as many factors as possible, within the constraints given, that is the most x could have in common with 2100.   50
<result>B</result>
Three is the largest number that can be divided evenly into 27 and the positive integer x, while 10 is the largest number that can be divided evenly into both 100 and x. Which of the following is the largest possible number that could be divided into x and 2500