Question 5

We are looking for numbers whose reciprocals have a repeating block of length one less than the number itself.

Step 1 — Understanding $1/7$

Let's look at the reciprocal of 7. We divide 1 by 7. $1 \div 7 = 0.142857142857...$ The repeating block is $\overline{142857}$. The length of this block is 6. This length is $7-1$.

$\boxed{\text{So, } \textbf{7} \text{ is a number with this property.}}$

Step 2 — Finding more numbers

We are looking for prime numbers $n$. Their reciprocal $1/n$ must have a repeating block. The length of this block must be $n-1$. These primes are called full reptend primes. Let's check some other prime numbers.

For 17: We divide 1 by 17. $1 \div 17 = 0.05882352941176470588...$ The repeating block is $\overline{0588235294117647}$. The length of this block is 16. This is $17-1$.

$\boxed{\text{So, } \textbf{17} \text{ is another such number.}}$

For 19: We divide 1 by 19. $1 \div 19 = 0.0526315789473684210526...$ The repeating block is $\overline{052631578947368421}$. The length of this block is 18. This is $19-1$.

$\boxed{\text{So, } \textbf{19} \text{ is another such number.}}$

For 23: We divide 1 by 23. $1 \div 23 = 0.04347826086956521739130434...$ The repeating block is $\overline{0434782608695652173913}$. The length of this block is 22. This is $23-1$.

$\boxed{\text{So, } \textbf{23} \text{ is another such number.}}$

For 29: We divide 1 by 29. $1 \div 29 = 0.03448275862068965517241379310344...$ The repeating block is $\overline{0344827586206896551724137931}$. The length of this block is 28. This is $29-1$.

$\boxed{\text{So, } \textbf{29} \text{ is another such number.}}$

Answer

Numbers whose reciprocals produce cyclic repeating blocks are prime numbers $n$ where the length of the repeating block is $n-1$. Examples of such numbers are: (i) 7 (ii) 17 (iii) 19 (iv) 23 (v) 29