Problem 17 - A Computational Introduction to Number Theory and Algebra

Question

Let \(n\) be a positive integer, and let \(X\) be a random variable, uniformly distributed over \(\\{0, \ldots, n-1\\} .\) For each positive divisor \(d\) of \(n\), let us define the random variable \(X_{d}:=X \bmod d\). Show that: (a) if \(d\) is a divisor of \(n,\) then the variable \(X_{d}\) is uniformly distributed over \(\\{0, \ldots, d-1\\}\) (b) if \(d_{1}, \ldots, d_{k}\) are divisors of \(n,\) then \(\left\\{X_{d_{i}}\right\\}_{i=1}^{k}\) is mutually independent if and only if \(\left\\{d_{i}\right\\}_{i=1}^{k}\) is pairwise relatively prime.

Step-by-Step Solution

Answer

Question: Prove that if a random variable X is uniformly distributed over the set {0, 1, 2, ..., n-1}, and d is a divisor of n, then the random variable X_d = X mod d is uniformly distributed over the set {0, 1, 2, ..., d-1}. Additionally, prove that if d_1, d_2, ..., d_k are divisors of n, then the set {X_d1, X_d2, ..., X_dk} is mutually independent if and only if the set {d_1, d_2, ..., d_k} is pairwise relatively prime.

Step 1: Proof of Statement A

First, we need to show that if \(d\) is a divisor of \(n\), the random variable \(X_d\) is uniformly distributed over \(\{0, \ldots , d-1\}\). Since \(X\) is a random variable uniformly distributed over \(\\{0, 1, 2,\ldots , n-1 \\}\), the probability that \(X\) takes on a specific value is \(\frac{1}{n}\), for each value in the given range. We have \(X_d = X \bmod d\). Let \(Y\) be another random variable in the given range that satisfies \(Y \equiv X \pmod{d}\). We need to show the probability of each integer \(i\) in the range \(\\{0, \ldots , d-1 \\}\) is also \(\frac{1}{d}\). Now let's consider \(\\{x | x \equiv i \pmod d \\}\). For each such \(x\), there exists a non-negative integer \(j\) such that \(x = i + jd\). As \(0 \leq x \leq n - 1\), there exists a maximum integer \(k\) such that \(i + kd \leq (n - 1)\), which gives \(k = \frac{n - i - 1}{d}\). Thus, there are \(k + 1\) numbers that are congruent to \(i\) modulo \(d\) in \(\\{0, 1, 2,\ldots , n-1 \\}\). Since \(X_d \equiv Y \pmod{d}\) and all values are equally likely, we have: \(\text{Pr}(X_d = i) = \frac{k+1}{n} = \frac{\frac{n-i-1}{d} + 1}{n} = \frac{1}{d}\). Hence, the variable \(X_d\) is uniformly distributed over \(\\{0, \ldots , d-1 \\}\).

Step 2: Proof of Statement B

Now we need to prove that if \(d_1, \ldots, d_k\) are divisors of \(n\), then \(\{X_{d_i}\}_{i=1}^k\) is mutually independent if and only if \(\{d_i\}_{i=1}^k\) is pairwise relatively prime. Let \(d_1, \ldots, d_k\) be pairwise relatively prime. Recall that random variables are mutually independent if the joint probability distribution is equal to the product of the individual probability distributions. We need to show that for any \(a_1, \ldots, a_k\): \(\text{Pr}(X_{d_1} = a_1, \ldots, X_{d_k} = a_k) = \text{Pr}(X_{d_1} = a_1)\cdot \ldots \cdot \text{Pr}(X_{d_k} = a_k)\). By the Chinese Remainder Theorem (CRT), since \(d_1, \ldots, d_k\) are pairwise relatively prime and \(0 \leq a_i \leq d_i - 1\), there exists a unique integer \(x \pmod{n}\) such that \(x \equiv a_i \pmod{d_i}\) for all \(i = 1, \ldots , k\). As proved earlier, for each \(i\), \(\text{Pr}(X_{d_i} = a_i) = \frac{1}{d_i}\). Now, we can show that \(\text{Pr}(X_{d_1} = a_1, \ldots, X_{d_k} = a_k) = \frac{1}{n}\). Since there is a unique integer \(x\) (modulo \(n\)) that satisfies the congruence conditions, we get: \(\text{Pr}(X_{d_1} = a_1, \ldots, X_{d_k} = a_k) = \frac{1}{n}\). This result holds for any value of \(a_1, \ldots, a_k\). Thus, we have: \(\text{Pr}(X_{d_1} = a_1, \ldots, X_{d_k} = a_k) = \text{Pr}(X_{d_1} = a_1)\cdot \ldots \cdot \text{Pr}(X_{d_k} = a_k)\). Now, we need to show that if \(\{X_{d_i}\}_{i=1}^k\) is mutually independent, then \(\{d_i\}_{i=1}^k\) is pairwise relatively prime. Suppose that \(d_i\) and \(d_j\) have a common divisor greater than 1, for some \(i \neq j\). If \(X_{d_i}\) and \(X_{d_j}\) are mutually independent, then all the congruence classes modulo \(d_i\) and \(d_j\) should be equally likely. However, since \(d_i\) and \(d_j\) are not relatively prime, there exists some congruence class that will never be reached simultaneously by \(X_{d_i}\) and \(X_{d_j}\), which contradicts the assumption that \(\{X_{d_i}\}_{i=1}^k\) is mutually independent. Thus, if \(\{X_{d_i}\}_{i=1}^k\) is mutually independent, then \(\{d_i\}_{i=1}^k\) is pairwise relatively prime. The proof is now complete.

Key Concepts

Random Variables

A random variable is a fundamental concept in probability theory, representing a variable whose possible values are numerical outcomes of a random phenomenon. Imagine rolling a die or flipping a coin – the results of these actions are random variables. In mathematics, each outcome of a random variable corresponds to a probability, showing how likely that outcome is.

There are two main types of random variables:

• Discrete Random Variables: These take on specific and defined values. For instance, the outcome when rolling a six-sided die can be 1, 2, 3, 4, 5, or 6.
• Continuous Random Variables: These can take on any value within a continuous range. A good example is measuring the height of people, which can vary smoothly over a range.

The concept of random variables is used to model real-world processes in various fields such as statistics, mathematics, engineering, and economics. In the context of Number Theory, random variables can help analyze properties like modular distributions and independence among variables linked by divisors.

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after reaching a certain value, called the modulus. It's like the clock arithmetic where after 12 o'clock, we go back to 1 o'clock. This system solves many practical problems by simplifying calculations using a remainder.

In modular arithmetic, you might often see expressions of the form \(a \equiv b \pmod{n}\), which means that when \(a\) is divided by \(n\), it leaves a remainder \(b\). Here, \(n\) is the modulus. This system is widely used in:

• Cryptography: Many encryption algorithms rely on modular arithmetic for security.
• Computer Science: Efficient algorithms often use modulo to ensure numbers stay within a range.
• Number Theory: Problems involving divisors and congruences use techniques of modular arithmetic.

In the context of the exercise, we examine \(X_d = X \bmod d\) to illustrate how values distribute when subjected to a modulus operation. The key insight is that this operation can make complex problems simpler by focusing only on remainders.

Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) is a pivotal result in number theory that deals with solving systems of simultaneous congruences. It shows how to reconstruct integers uniquely from given remainders for several pairwise coprime modulus. This theorem provides a powerful method to simplify a problem involving several congruences into a more manageable form.

The CRT states that if you have several integers \(n_1, n_2, \ldots, n_k\) that are pairwise coprime (the greatest common divisor of each pair is 1), and arbitrary integers \(a_1, a_2, \ldots, a_k\), then there is a unique integer \(x\) modulo \(N = n_1 n_2 \ldots n_k\), such that:

• \(x \equiv a_1 \pmod{n_1}\)
• \(x \equiv a_2 \pmod{n_2}\)
• \(\cdots\)
• \(x \equiv a_k \pmod{n_k}\)

The CRT is essential in ensuring that random variables \(X_{d_i}\) in the problem are mutually independent. Because, if the moduli \(d_1, \ldots, d_k\) are pairwise relatively prime, each sequence can be treated separately, making calculations easier and more structured.