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TMUA 2023 · Paper 2 · Question 15 of 20

TMUA 2023 Paper 2 Question 15

Number and divisibility — Recurring numbers in base 2. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 2Number and divisibilityRecurring numbers in base 28 options
The base 10 number 0.03841 has the value 0× 10−1 + 3× 10−2 + 8× 10−3 + 4× 10−4 + 1× 10−5 = 0.03841 Similarly, the base 2 number 0.01101 has the value 0× 2−1 + 1× 2−2 + 1× 2−3 + 0× 2−4 + 1× 2−5 = 1332 What is the value of the recurring base 2 number 0.0011 = 0.001100110011…?
  1. A13
  2. B15
  3. C115
  4. D215
  5. E415
  6. F316
  7. G516
  8. H631
Show the answer and worked solution
answer · B
  1. A13
  2. B15
  3. C115
  4. D215
  5. E415
  6. F316
  7. G516
  8. H631
The block 0011 repeats with period 4. One block contributes 0⋅ 2−1 + 0⋅ 2−2 + 1⋅ 2−3 + 1⋅ 2−4 = 18 + 116 = 316, and each later block is 2−4 = 116 times the one before. Summing the geometric series, 3/161 − 1/16 = 3/1615/16 = 315 = 15.