Math 365 Lecture Notes - Divisibility and Integers

A set of flashcards based on key concepts from Math 365 lecture notes on divisibility and integers.

What is the definition of divisibility for whole numbers n and d?

A whole number d divides n, if and only if there exists a unique whole number q such that n = dq.

What is an even number?

A whole number that is divisible by 2, meaning the remainder is 0 when divided by 2.

What is an odd number?

A whole number that is not divisible by 2, meaning the remainder is 1 when divided by 2.

According to Theorem 4-1, if d|a, what else can be inferred?

If d divides a, then d divides every multiple of a.

What is a prime number?

Whole numbers that have exactly two distinct whole-number divisors.

What is a composite number?

Whole numbers that have more than two distinct whole-number divisors.

What is the Greatest Common Divisor (GCD)?

The largest whole number that divides both a and b.

What is the Least Common Multiple (LCM)?

The smallest natural number that is both a multiple of a and a multiple of b.

What is necessary to check divisibility by 9?

The sum of the digits of the number must be divisible by 9.

How do you determine if a number is divisible by 12?

A number is divisible by 12 if it is divisible by both 3 and 4.

What defines two numbers as relatively prime?

Their greatest common divisor (GCD) is 1.

What is a unique characteristic of the number 2 among prime numbers?

It is the only even prime number.

What is the relationship between divisibility by 2, 3, and 6?

A number is divisible by 6 if it is divisible by both 2 and 3.

What does the Fundamental Theorem of Arithmetic state?

The prime factorization of a number is unique, except for the order of the factors.

How do you classify the number 1 in terms of primality?

It is neither prime nor composite.

What is necessary for divisibility by 4?

The number formed by the two rightmost digits must be divisible by 4.

What is the significance of the Sieve of Eratosthenes?

It is a method for finding all prime numbers up to a specified integer.

How is divisibility by 5 determined?

A number is divisible by 5 if its units digit is 0 or 5.