Problem 23 - A Computational Introduction to Number Theory and Algebra

Question

Let \(z=s / t \in \mathbb{Q},\) where \(s\) and \(t\) are relatively prime integers with \(0 \leq s

Step-by-Step Solution

Answer

In this exercise, we proved multiple properties about the decimal expansion of a given fraction \(z=s/t\), where \(s\) and \(t\) are relatively prime integers, using modular arithmetic and properties of relatively prime integers. We established the existence of integers \(k, k'\) such that \(s10^k \equiv s10^{k'} \pmod{t}\) using the Pigeonhole Principle. Then, we proved that the decimal expansion of \(z\) is periodic if and only if the congruence holds. Furthermore, we showed that the decimal expansion of \(z\) is purely periodic with a period equal to the multiplicative order of 10 modulo \(t\) under certain conditions. Finally, we established that the decimal expansion of \(z\) is ultimately periodic with a pre-period \(k\) and a period equal to the multiplicative order of 10 modulo \(t'\).

Step 1: Part (a): Existence of integers \(k, k'\) with given properties

Let's first consider the sequence of integers \(s10^0, s10^1, s10^2, \ldots, s10^{t-1}\). By the Pigeonhole Principle, there must be two numbers in this sequence with the same remainder when divided by \(t\). Let these integers be \(s10^k\) and \(s10^{k'}\), with \(0 \leq k

Step 2: Part (b): Decimal expansion of \(z\) is periodic if and only if the congruence holds

To prove this part, we need to show that the decimal expansion of \(z\) is \((k, k'-k)\)-periodic if and only if \(s 10^k \equiv s 10^{k'} \pmod{t}\) holds: (i) If the decimal expansion of \(z\) is \((k, k'-k)\)-periodic, then we can write \(z\) as: \(z= x.\overline{c_{k+1}c_{k+2}\ldots c_{k'-1}c_{k'}}\) where \(x\) and the \(c_i\)s are some non-negative integers. Multiplying both sides by \(10^{k'-k}\), the decimal expansion of \(z10^{k'-k}\) will be: \(z 10^{k'-k} = x10^{k'-k} + \overline{c_{k+1}c_{k+2}\ldots c_{k'-1}c_{k'}}\) Since \(z 10^{k'-k} - z = x10^{k'-k} - x\) is an integer, we have: \(z(10^{k'-k} - 1) \equiv 0 \pmod{t}\) Multiplying by \(10^k\): \(z10^{k'} - z10^k \equiv 0 \pmod{t}\) Hence: \(s 10^k \equiv s 10^{k'} \pmod{t}\) (ii) Conversely, let's assume we have \(s 10^k \equiv s 10^{k'} \pmod{t}\). Then, we can write: \(s10^{k'} - s10^k = rt\) for some integer \(r\). Dividing both sides by \(s\): \(10^{k'} - 10^k = rt/s\) And: \(\frac{10^{k'} - 10^k}{10^k} \cdot \frac{s}{t} = r\) Since the left side is an integer, the decimal expansion of \(z\) has a period of length \(k'-k\) preceded by \(k\) non-repeating decimal places. Thus, the decimal expansion of \(z\) is \((k, k'-k)\)-periodic. This proves part (b).

Step 3: Part (c): Decimal expansion of \(z\) is purely periodic with period equal to the multiplicative order of 10 modulo \(t\)

Assume that \(\gcd(10, t) = 1\). Let \(n\) be the multiplicative order of 10 modulo \(t\). Then, \(10^n \equiv 1 \pmod{t}\) and \(10^j \not\equiv 1 \pmod{t}\) for any \(0 < j < n\). We now have: \(z10^n - z\equiv 0 \pmod{t}\), since \(10^n \equiv 1 \pmod{t}\). Thus, \(s10^n - s \equiv 0 \pmod{t}\). Also, \(s 10^k \not\equiv s \pmod{t}\) for all \(0 < k < n\), since \(\text{gcd}(10, t) = 1\). Using the result from part (b), we conclude that the decimal expansion of \(z\) is purely periodic with period equal to the multiplicative order of 10 modulo \(t\), which is \(n\). This proves part (c).

Step 4: Part (d): The decimal expansion of \(z\) is ultimately periodic with pre-period \(k\) and period equal to the multiplicative order of 10 modulo \(t'\)

Let \(k\) be the smallest non-negative integer such that 10 and \(t'=t/\gcd(10^k, t)\) are relatively prime. Then, we can write \(t=2^a5^bt'\) with \(a, b \ge 0\) and \(\text{gcd}(10, t') = 1\). Let \(n\) be the multiplicative order of 10 modulo \(t'\). Then, \(10^{k+n} - 10^k \equiv 0 \pmod{t'}\) and \(10^j - 10^k \not\equiv 0 \pmod{t'}\) for any \(0 < j < k+n\). We now have: \(z(10^{k+n} - 10^k) \equiv 0 \pmod{t}\), since \(10^{k+n} - 10^k \equiv 0 \pmod{t'}\). Thus, \(s(10^{k+n} - 10^k) \equiv 0 \pmod{t}\). Using the result from part (b), we conclude that the decimal expansion of \(z\) is ultimately periodic with pre-period \(k\) and period equal to the multiplicative order of 10 modulo \(t'\), which is \(n\). This proves part (d).

Key Concepts

Congruences

In number theory, congruences are a way to express that two numbers leave the same remainder when divided by a given number, often called the modulus. Congruences are written in the form \(a \equiv b \pmod{m}\), which reads as 'a is congruent to b modulo m'. This tells us that when both \(a\) and \(b\) are divided by \(m\), they result in the same remainder.

Congruences play a critical role in number theory and modular arithmetic. For example, consider two powers of a number multiplied by the same integer (like \(s10^k\) and \(s10^{k'}\) from the exercise). By finding a congruence such as \(s10^k \equiv s10^{k'} \pmod{t}\), we essentially state that both expressions have the same remainder when divided by \(t\).

This provides a method to determine periodic patterns in decimal expansions of fractions, a tool that deepens our understanding of the properties of rational numbers. Understanding congruences allows us to predict these patterns and find systematic solutions in many numerical problems.

Decimal Expansion

Decimal expansion refers to the way in which a fraction or a number is expressed in base 10, our standard numeral system. Rational numbers, which can be expressed as the quotient of two integers, have decimal expansions that either terminate or repeat eventually.

The type of decimal expansion a rational number shows depends primarily on the denominator of its simplest form. If the denominator of a simplified fraction can only be factored into the powers of 2 and 5, the decimal will terminate. Otherwise, it will result in a repeating or periodic decimal expansion.

In the context of the exercise, the rational number \(z = s / t\) where \(s\) and \(t\) are coprime, can have a decimal expansion that is periodic. This periodicity links directly to the congruences examined earlier. Understanding this helps to predict when and how decimal places repeat, based on calculations involving the modulus of the denominator.

Periodic Sequences

Periodic sequences are sets of numbers where the sequence follows a repeating pattern after a certain number of terms. In the context of decimal expansions, a periodic sequence implies that there is a repeating block of numbers in the decimal form of a fraction.

The determination of repeats in a decimal expansion can be revealed through the analysis of congruences. If, as in the exercise, \(s10^k \equiv s10^{k'} \pmod{t}\), this relationship indicates a repeat in the decimal sequence starting at position \(k\) and wrapping back after \(k'-k\) digits.

Periodic sequences have applications in solving real-world problems where consistency and predictability in numeric flows are crucial. Recognizing and understanding these sequences help in areas like cryptography and computing, where repetition can be both a tool and a challenge.

Multiplicative Order

The multiplicative order of an integer with respect to a modulus is the smallest power to which the integer can be raised to yield a product that is congruent to one modulo a given integer. For example, in the exercise where \( \gcd(10, t) = 1 \), we seek the smallest integer \(n\) such that \(10^n \equiv 1 \pmod{t}\).

This concept is crucial in the study of periodic decimal expansions, as it directly determines the period of repetition. If \(n\) is the multiplicative order, then our decimal sequence corresponding to the fraction \(s/t\) will have a block of numbers repeating every \(n\) digits.

Understanding the multiplicative order helps in predicting the length of the periodic section of decimal expansions and is instrumental in various applications including coding theory, where predictable cyclic patterns are necessary for error correction and data encryption.

Previous

Problem 22

Next

Problem 24

Other exercises in this chapter

Problem 21

Show that if a sequence \(\Psi\) is ultimately periodic, then it is \(\left(k^{*}, \ell^{*}\right)\) -periodic for some uniquely determined pair \(\left(k^{*},

View solution

Problem 22

Let \(z\) be a real number whose decimal expansion is an ultimately periodic sequence. Show that \(z\) is rational.

View solution

Problem 24

This exercise develops a method to speed up RSA decryption. Suppose that we are given two distinct \(\ell\) -bit primes, \(p\) and \(q,\) an element \(\beta \in

View solution

Problem 25

Alice submits a bid to an auction, and so that other bidders cannot see her bid, she encrypts it under the public key of the auction service. Suppose that the a

View solution

Key Concepts

Congruences

In number theory, congruences are a way to express that two numbers leave the same remainder when divided by a given number, often called the modulus. Congruences are written in the form \(a \equiv b \pmod{m}\), which reads as 'a is congruent to b modulo m'. This tells us that when both \(a\) and \(b\) are divided by \(m\), they result in the same remainder.

Congruences play a critical role in number theory and modular arithmetic. For example, consider two powers of a number multiplied by the same integer (like \(s10^k\) and \(s10^{k'}\) from the exercise). By finding a congruence such as \(s10^k \equiv s10^{k'} \pmod{t}\), we essentially state that both expressions have the same remainder when divided by \(t\).

This provides a method to determine periodic patterns in decimal expansions of fractions, a tool that deepens our understanding of the properties of rational numbers. Understanding congruences allows us to predict these patterns and find systematic solutions in many numerical problems.

Decimal Expansion

Decimal expansion refers to the way in which a fraction or a number is expressed in base 10, our standard numeral system. Rational numbers, which can be expressed as the quotient of two integers, have decimal expansions that either terminate or repeat eventually.

The type of decimal expansion a rational number shows depends primarily on the denominator of its simplest form. If the denominator of a simplified fraction can only be factored into the powers of 2 and 5, the decimal will terminate. Otherwise, it will result in a repeating or periodic decimal expansion.

In the context of the exercise, the rational number \(z = s / t\) where \(s\) and \(t\) are coprime, can have a decimal expansion that is periodic. This periodicity links directly to the congruences examined earlier. Understanding this helps to predict when and how decimal places repeat, based on calculations involving the modulus of the denominator.

Periodic Sequences

Periodic sequences are sets of numbers where the sequence follows a repeating pattern after a certain number of terms. In the context of decimal expansions, a periodic sequence implies that there is a repeating block of numbers in the decimal form of a fraction.

The determination of repeats in a decimal expansion can be revealed through the analysis of congruences. If, as in the exercise, \(s10^k \equiv s10^{k'} \pmod{t}\), this relationship indicates a repeat in the decimal sequence starting at position \(k\) and wrapping back after \(k'-k\) digits.

Periodic sequences have applications in solving real-world problems where consistency and predictability in numeric flows are crucial. Recognizing and understanding these sequences help in areas like cryptography and computing, where repetition can be both a tool and a challenge.

Multiplicative Order

The multiplicative order of an integer with respect to a modulus is the smallest power to which the integer can be raised to yield a product that is congruent to one modulo a given integer. For example, in the exercise where \( \gcd(10, t) = 1 \), we seek the smallest integer \(n\) such that \(10^n \equiv 1 \pmod{t}\).

This concept is crucial in the study of periodic decimal expansions, as it directly determines the period of repetition. If \(n\) is the multiplicative order, then our decimal sequence corresponding to the fraction \(s/t\) will have a block of numbers repeating every \(n\) digits.

Understanding the multiplicative order helps in predicting the length of the periodic section of decimal expansions and is instrumental in various applications including coding theory, where predictable cyclic patterns are necessary for error correction and data encryption.