Welcome back to this week’s Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 minutes and go!
What is the 18th digit to the right of the decimal point in the decimal expansion of 1/37?
There is no way to intuit the answer; we must simply divide 1 by 37. However, we can be certain that we will be able to spot a pattern. The GMAT would not require us to compute all the way to the 18th digit. Moreover, every fraction with integers on top and bottom can be expanded into a decimal that either terminates (e.g., 1/4 = 0.25) or repeats. For instance, 1/3 = 0.333…, with an infinite series of repeating 3’s. Likewise, 1/11 = 0.090909…, with an infinite series of 09’s.
Since the denominator of the fraction given in the problem is 37, we know that the fraction will repeat. (For a fraction to terminate, the denominator, after reducing, must have only 2’s or 5’s or both as factors.) So we should do the long division, looking for the repeating cycle.
Perform the long division. You will probably have to experiment with the multiples of 37 to discover that 7 × 37 = 259.
Once we get back to a remainder of 1, then we know the cycle will start all over again. Therefore, 1/37 = 0.027027…, with a repeating cycle of 027. This cycle contains 3 digits, so every third digit will be the same. For instance, 7 will be the 3rd digit, the 6th digit, the 9th digit, and so forth in the decimal expansion. Since 18 is also a multiple of 3 (like 3, 6, and 9), the digit 7 will be in the 18th position as well.
The correct answer is (D) 7.
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