Introduction

The problem of determining the length of the smallest repeating block in the decimal expansion of 5/7 is an interesting and fundamental concept in number theory. This exploration will help us understand the intricacies of repeating decimals and their significance in various mathematical contexts.

Understanding the Multiplicative Order

To answer the question of how many digits does the smallest repeating block in the decimal expansion of 5/7 contain, we need to delve into the concept of the multiplicative order. The multiplicative order of a number 7 with respect to 10 can be defined as the smallest positive integer n such that:

10^n ≡ 1 (mod 7)

First, we need to ensure that 7 and 10 are co-prime, meaning their greatest common divisor (GCD) is 1. In this case, 7 and 10 are indeed co-prime, so we can proceed with our analysis. To find the smallest value of n that satisfies the condition:

10^n ≡ 1 (mod 7)

We can test successive values of n:

n 1: 10^1 ≡ 3 (mod 7) n 2: 10^2 ≡ 2 (mod 7) n 3: 10^3 ≡ 6 (mod 7) n 4: 10^4 ≡ 4 (mod 7) n 5: 10^5 ≡ 5 (mod 7) n 6: 10^6 ≡ 1 (mod 7)

From the above calculations, we observe that the smallest n that satisfies the condition 10^n ≡ 1 (mod 7) is 6. Therefore, the smallest repeating block in the decimal expansion of 5/7 contains 6 digits.

Repeating Block in Rational Numbers

For any rational number of the form x/7, the decimal expansion will either terminate with a repeating sequence or be entirely repeating, depending on the value of x. Specifically, if x is a number between 1 and 6 (inclusive), the decimal expansion will have a repeating sequence without any non-repeating prefix.

The repeating unit for any multiple of 1/7 will always consist of 6 digits. This is true regardless of the base in which the number is expressed. To illustrate, consider the following examples:

1/7 0.142857 2/7 0.285714 3/7 0.428571 4/7 0.571428 5/7 0.714285 6/7 0.857142

Each of these fractions has a repeating sequence of 6 digits. Additionally, note that the digits from one fraction can be permuted to form the repeating sequence of another fraction, as mentioned in the example where 5/7 0.714285 and others are derived by cyclic shifts of the digits 142857.

Repeating Units in 5/7

Let's consider the decimal expansion of 5/7 more closely. The decimal representation of 5/7 is:

0.714285714285714285...

The repeating unit here is 714285, which consists of 6 digits. This pattern repeats infinitely, indicating that the length of the smallest repeating block is indeed 6 digits.

Conclusion

In summary, the decimal expansion of 5/7 has a repeating block of 6 digits. The concept of the multiplicative order of 7 with respect to 10 helps us determine this length. This property is not unique to 7; multiples of other numbers with similar properties will also exhibit repeating blocks of the same length. Understanding this concept is crucial for students and researchers working in number theory and related fields.