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If you divide four by seven, what is the thirty-seventh digit to the right of the decimal point of the resulting number?
Correct Answer: D
Find a pattern in the decimal digits of the division operation.
  1. The first step is to realize that we do not have to divide 4 by 7 to thirty seven places to find this answer. Because the test writers would not test such tedious math, there must be a pattern here, which would consist of a repeating decimal.
  2. Therefore this question tests our ability to a) do long division and b) recognize a pattern with a repeating decimal.
  3. Setting up this long division problem, work this problem out until you notice the same number twice.
    Figure 1
  4. Notice, that, after we multiplied 8 by 7, to get 56, we have 4 left over. Bringing down the 0, we next have to divide 40 by 7. This, however, is the same division problem that we started with. The following numbers in the answer must therefore be 5, 7, 1, etc.
  5. Therefore, this decimal repeats every 6 places, with the pattern of 571428. We are looking for the number that will be in the 37th place, so we divide 37 by 6 (the number of numbers in the pattern) to get 6 with a remainder of 1. This means that there will be 6 complete sets of our repeating pattern (571428), and that the first digit of the set (the remainder) must be the 37th digit.
  6. Thus, the answer is our first number of the pattern, 5, answer choice (D).

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