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Chapter 4: Problem 31

Show that a positive integer is divisible by 3 if and only if the sum of its decimal digits is divisible by \(3 .\)

Short Answer

Expert verified
An integer is divisible by 3 if the sum of its digits is divisible by 3.

Step by step solution

01

- Understand the Problem

The goal is to show that a positive integer is divisible by 3 if and only if the sum of its digits is also divisible by 3.
02

- Represent the Integer

Represent a positive integer in its decimal form. For example, the integer can be written as: \[ n = a_k \times 10^k + a_{k-1} \times 10^{k-1} + \text{...} + a_1 \times 10^1 + a_0 \] where \(a_k, a_{k-1}, ..., a_1, a_0\) are the decimal digits.
03

- Consider Modulo 3

To check for divisibility by 3, consider the properties of modulus 3. Note that: \[ 10 \bmod 3 = 1 \] since \(10\) divided by 3 leaves a remainder of 1.
04

- Simplify under Modulo 3

Given the expression for the integer, rewrite it considering modulo 3: \[ n \bmod 3 = (a_k \times 10^k + a_{k-1} \times 10^{k-1} + ... + a_1 \times 10 + a_0) \bmod 3 \] can be simplified as: \[ n \bmod 3 = (a_k \times (10 \bmod 3)^k + a_{k-1} \times (10 \bmod 3)^{k-1} + ... + a_1 \times (10 \bmod 3) + a_0) \bmod 3 \]
05

- Compute Using Modulo Properties

Since \( 10 \bmod 3 = 1 \), the equation becomes: \[ n \bmod 3 = (a_k \times 1^k + a_{k-1} \times 1^{k-1} + ... + a_1 \times 1 + a_0) \bmod 3 \] which simplifies to: \[ n \bmod 3 = (a_k + a_{k-1} + ... + a_1 + a_0) \bmod 3 \]
06

- Conclude the Proof

Thus, we have: \[ n \bmod 3 = (a_k + a_{k-1} + ... + a_1 + a_0) \bmod 3 \] This means that the integer \(n\) is divisible by 3 if and only if the sum of its decimal digits is divisible by 3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decimal Representation
To understand the problem, first, we need to break down the integer into its decimal representation. Decimal representation is when a number is expressed in the base 10 system, which we use in everyday arithmetic. For example, the number 1234 can be written as: \(1234 = 1 \times 10^3 + 2 \times 10^2 + 3 \times 10^1 + 4 \times 10^0\)
Here:
  • 1, 2, 3, 4 are the decimal digits.
  • Each digit is multiplied by a power of 10.
Understanding this helps us to see how changing the digits changes the value of the number. Every number can be represented in this way using its digits and powers of 10.
Modulo Operation
The modulo operation is crucial in number theory. It finds the remainder when one integer is divided by another. For example, \(10 \bmod 3\) finds the remainder when 10 is divided by 3, which is 1. Mathematically, \(10 \bmod 3\ = 1\).
Properties of modulo operation help simplify expressions significantly. In our exercise, by knowing \(10 \bmod 3 = 1\), we can simplify the complex expression involving powers of 10. For any \(10^k \bmod 3\), it simplifies to \(1^k \bmod 3 = 1\). This property allows us to reduce a potentially large and complicated equation to a much simpler form.
Number Theory
Number theory is a branch of pure mathematics dedicated to the study of integers. It's deeply connected with concepts such as divisibility, prime numbers, and modular arithmetic. In this exercise, we focus on the concept of divisibility. A number is divisible by another if there is no remainder. For example, \(9 \div 3 = 3\), so 9 is divisible by 3.
In our problem, we show that checking whether a number is divisible by 3 is equivalent to checking if the sum of its decimal digits is divisible by 3. This relationship simplifies checking for divisibility without performing division directly. It is a powerful shortcut that leverages our understanding of decimal representation and the properties of modulo operation. This is a classic example of how number theory provides elegant solutions to seemingly complex problems.

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