philipssonicare wrote:

We should memorise division by which numbers? 3, 7, 11, any others?

Please address @Yashbati 's concern as it is a good point

The question in the OP isn't a realistic GMAT problem (partly because to confirm the right answer, E, you need long division). So I wouldn't worry about it much, but PKN's method is about as fast as you can get here.

I cannot imagine a GMAT situation where there would be any advantage to knowing the exact decimal equivalent of 1/7 (which is a six-digit repeating pattern). It should be enough to know that it's roughly 0.14. It is occasionally useful to know the exact decimals of 1/9 and 1/11 though (and how to find related decimal expansions, like 3/11 or 1/99).

yashbhati wrote:

If we have 2 primes in the denominator ( other than 2 and 5) then the number of repeating decimals depends on which number? Is the larger prime or the prime with more repeating decimals?

The theory behind this is miles beyond the GMAT, and you'll never need to know about it for the test. But if you do have a fraction 1/(pq), where p and q are different primes not equal to 2 or 5, then if the decimal equivalent of 1/p repeats in a pattern that is D digits long, and the decimal equivalent of 1/q repeats in a pattern that is E digits long, the decimal of 1/(pq) will repeat in a pattern that is LCM(D, E) (the least common multiple of D and E) digits long.

So for example, since 1/11 repeats in a pattern 2 digits long, and 1/111 repeats in a pattern 3 digits long, 1/(11)(111) = 1/1221 would repeat in a pattern 6 digits long, because 6 is the LCM of 2 and 3. Or, as another example, 1/(7)(11) = 1/77 will repeat in a pattern 6 digits long, because 1/7 has a 6-digit pattern, and 1/11 has a 2-digit pattern, and 6 is the LCM of 6 and 2. You can confirm those results with any calculator.

You'll never need that on the GMAT though.

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