If k is a common multiple of 75, 98, and 140, which of the following statements are true?
I. k is divisible by 9
II. k is divisible by 49
III. k is greater than 14,000
A. II only
B. III only
C. I and II only
D. II and III only
E. I, II, and III
0 is a multiple of each of the three numbers, as is -14,700, etc. The Least Common Multiple of three positive integers is defined as the least positive common multiple because there is no least common multiple otherwise: any arbitrarily small negative number will do.
It seems controversial to declare that all common multiples must be greater than or equal to the Least Common Multiple when the LCM isn't actually the "least" one, but is simply named that and defined otherwise. (Not to make a bad pun, but it ought to be called the Least Natural Common Multiple.) It seems even more controversial to punish test takers with a different notion of multiples for selecting answer A, especially given the GMAT's propensity to test students' ability to consider prominent exceptional numbers, such as 0.
What's your source for your interpretation? In his "Elementary Theory of Numbers", Sierpinski makes the distinction between "common multiples" and "common multiples which are natural numbers", and he explicitly defines the LCM as the smallest of the second set. I'm curious what would supersede this.
