Question 42 of 58advanced🔍 AnalyzeEssay3 marks

We have seen that the repeating block of 17\frac{1}{7} is a cyclic number. Try to find more numbers (n)(n) whose reciprocals (1n)\left(\frac{1}{n}\right) produce decimals with repeating blocks that are cyclic.

Correct Answer

Another number whose reciprocal produces a decimal with a cyclic repeating block is n=17n = 17. Calculating 117\frac{1}{17} reveals a specific repeating block of digits.

Similar to the repeating block of 17\frac{1}{7} (142857), the digits for 117\frac{1}{17} show cyclic properties. When we evaluate multiples like 217,317\frac{2}{17}, \frac{3}{17}, and so on, the same digits shift in a cyclic circle rather than changing randomly.

This demonstrates that 7 is not unique; 17 also generates a cyclic number. These numbers highlight the beautiful and symmetrical internal patterns hidden within certain rational numbers.

Exercise: Exercise Set 3.5 | Q: *5 | (Page 22)
For More Understanding

Explanation

A cyclic number is generated by a full reptend prime pp, where the period of 1/p1/p equals p−1p-1. For 7, the period is 6. The next such prime is 17, where the period of 1/17 is 16. The repeating block of 1/17 is 0588235294117647, and multiplying this block by 2, 3, ..., 16 produces cyclic permutations of the same digits, just as 142857 does for 1/7. Other examples include 19, 23, and 29.

Solution Steps

  1. Step 1: Identify that a cyclic number is generated by a full reptend prime pp, where the period of 1/p1/p equals p−1p-1.

  2. Step 2: Test the next prime numbers after 7. For n=13n=13, the period of 1/131/13 is 6, not 12, so it is not a full reptend prime.

  3. Step 3: Test n=17n=17. The period of 1/171/17 is 16, which equals 17−117-1. The repeating block is 0588235294117647.

  4. Step 4: Explain that just like 1/7, the fractions 2/17, 3/17, etc., show cyclic properties where the same digits shift in a cyclic circle.

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