gmatwithpooja wrote:
why it is not q = 166*a+1 where a is any integer....just like for 13, 1/13 has a repeating period of 6 so 13 = 6*2+1
also, will this be a real GMAT question. i have honest doubts!
Dear
gmatwithpooja,
I'm happy to respond.
As a general rule, the patterns that prime numbers follow are in no way amenable to simple algebra formula. In fact, the ultimate pattern of prime numbers is the single hardest unsolved problem in modern mathematics, known as the
Riemann Hypothesis. This what folks with Ph.D.'s in pure mathematics try to tackle. This is leagues and leagues beyond what you need to know for the GMAT. The only rough-and-ready fact that you should know is that no simple algebra formula in the universe is going to predict how prime numbers behave.
One of the behaviors of prime numbers concerns how many repeating digits are in the decimal representation of their reciprocals. For example
1/3 = 0.3333333... (a repeating pattern one digit long)
1/7 = 0.142857 142857 142857 142857 142857 ... (a repeating pattern 6 digits long)
1/11 = 0.09090909090909... (a repeating pattern 2 digits long)
1/13 = 0.076923 076923 076923 076923 076923 ... (a repeating pattern 6 digits long)
1/17 = 0.0588235294117647 0588235294117647 ... (a repeating pattern 16 digits long)
The pattern is: for prime number P, the decimal representation of the reciprocal could have (P - 1) digits, which would be the maximal possible value, or it could have a number of repeating digits that is any factor of (P - 1).
3 - 1 = 2, and 1 is a factor of 2
7 - 1 = 6, the maximum value
11 - 1 = 10, and 2 is a factor of 10
13 - 1 = 12, and 6 is a factor of 12
17 - 1 = 16, the maximum value
There is no easy algebraic pattern that relates the number of repeating digits to P. Some prime numbers have the maximum value of repeating digits, and others don't. It's very idiosyncratic. All of this is also well beyond what you need for the GMAT.
Again, the one concrete take-away you need is: nothing about prime numbers can be predicted or understood with a simple algebraic formula.
For more information on decimals, including facts you do need for the GMAT, see this blog article:
GMAT Math: Terminating and Repeating DecimalsDoes all this make sense?
Mike
_________________
Mike McGarry
Magoosh Test PrepEducation is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)