If a is a positive integer and \(\frac{22a}{210}\) is also an integer, how many different positive prime factors could a possibly have at most given that a is less than 150?

A) Two
B) Three
C) Four
D) Five
E) Six
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Let’s simplify the given expression:

22a/210 = 11a/105

Thus, we see that a/105 is an integer. In other words, a = 105k for some positive integer k. However, since we are given that a is less than 150, we see that k must be 1 and thus a must be 105.

Since 105 = 7 x 3 x 5, we see that a could have 3 different prime factors.

Answer: B
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