We have seen that the repeating block of 17 is a cyclic number. Try to find more numbers (n) whose reciprocals (1n) produce decimals with repeating blocks that are cyclic.
Step-by-step solution
Idea: A block is cyclic when 2n, 3n, …, n − 1n are all rotations of the block of 1n. That happens when the long division of 1 by n passes through every remainder 1, 2, …, n − 1 before repeating, i.e. the block has n − 1 digits.
- For 17 the remainders run through all of 1–6, so the block 142857 has 7 − 1 = 6 digits and every k7 starts at a different point of the same loop. Look for other n whose block has n − 1 digits.1 mark
- Test small primes: 13 = 0.3 (1 digit), 111 = 0.09 (2 digits), 113 = 0.076923 (6 digits): none has n − 1 digits. 117 = 0.0588235294117647 has 16 = 17 − 1 digits ✓.1 mark
- Check 17: 217 = 0.1176470588235294 and 317 = 0.1764705882352941, both rotations of 0588235294117647. Similarly 119 = 0.052631578947368421 (18 digits) is cyclic. Others: 23, 29, 47, 59, 61, 97.1 mark
Check: 0588235294117647 × 17 = 9999999999999999 (sixteen 9s) ✓; 052631578947368421 × 19 = 999999999999999999 ✓.
Answer to write in the exam
Cyclic ⇔ repeating block of 1n has n − 1 digits
117 = 0.0588235294117647 (16 digits)
217 = 0.1176470588235294 (rotation)
119 = 0.052631578947368421 (18 digits)
∴ n = 17, 19, 23, 29 (also 47, 59, 61, 97) give cyclic blocks.
Common mistakes that cost marks
- Picking 13 because 1/13 repeats: its block 076923 has only 6 digits, and 213 is not a rotation of it.
- Dropping the leading 0 of the block for 117 (0588…); the rotation property only works with all 16 digits.
- Choosing a non-prime such as 21 or 49; the full-length behaviour needs a prime (other than 2 and 5).
How this can come in the exam
The repeating block of 117 has how many digits?
- 8
- 15
- 16
- 17
Show answer
(C) 16
17 − 1 = 16 digits: 0588235294117647.
Given 17 = 0.142857, write 57 without dividing and explain how.
Show answer
142857 × 5 = 714285, a rotation of the block (1 mark). So 57 = 0.714285 (1 mark).Try one yourself
Is the repeating block of 111 cyclic? Explain.
Show answer
No. 111 = 0.09 has 2 digits, not 10; e.g. 211 = 0.18 is not a rotation of 09.
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