Sri SaiRam engineering College
Introduction to number theory lecture 3: Divisibility and Euclid's algorithms.
Introduction
In this lecture, we will discuss the basic properties of divisibility, specifically showing that 6
divides n*(n+1)*(n+2) for any integer n.
Divisibility
We can show that 6 divides n*(n+1)*(n+2) for any integer n by noting that n must be of the form
of 3n, 3n+1, or 3n+2. In any of these three cases, one of the factors is divisible by 3, so the
entire expression is divisible by 3. Additionally, since 2 and 3 are coprime, this implies that the
expression is also divisible by 2, hence divisible by 6. A much neater way to show this is to
identify it with the binomial coefficient n+2 choose 3. To prove this, we identify it with the
binomial coefficients n+1 choose 2 or n+2 choose 3, which are always integers, and then show
that it is divisible by 6.
Ideals
The set of x such that d divides x is something called an ideal of the integers. An ideal is just
something that's closed under addition and subtraction.
Euclid's Division Algorithm
The algorithm is the number of times you have to take the remainder to subtract a, denoted by r,
will be the number to subtract and the remainder r. We also have a quotient. We repeat this
process until there are no more copies of a left to subtract from b. If a is negative, you wouldn't
add copies, but instead subtract one. If b is negative, you would have to add copies of b or a to
subtract a from b. If both a and b are positive, you would subtract a from b.
The Greatest Common Divisor and Euclid's Algorithm
The greatest common divisor (GCD) is the largest integer that divides two numbers without
leaving a remainder. We can use Euclid's division algorithm to find the GCD:
Suppose we have two numbers, a and b.
If a or b is zero, the GCD is the other number.
Otherwise, we find the largest integer d such that d divides both a and b.
However, finding the GCD of large numbers can be difficult. Euclid's algorithm is a more efficient
method:
Factor a and b into primes.
Find the product of all common prime factors, raised to the minimum power.
The result is the GCD.
While Euclid's algorithm is still difficult for extremely large numbers, it is widely used in modern
computing. For example, it is used to solve equations and to factor large numbers.
The Greatest Common Divisor
Introduction to number theory lecture 3: Divisibility and Euclid's algorithms.
Introduction
In this lecture, we will discuss the basic properties of divisibility, specifically showing that 6
divides n*(n+1)*(n+2) for any integer n.
Divisibility
We can show that 6 divides n*(n+1)*(n+2) for any integer n by noting that n must be of the form
of 3n, 3n+1, or 3n+2. In any of these three cases, one of the factors is divisible by 3, so the
entire expression is divisible by 3. Additionally, since 2 and 3 are coprime, this implies that the
expression is also divisible by 2, hence divisible by 6. A much neater way to show this is to
identify it with the binomial coefficient n+2 choose 3. To prove this, we identify it with the
binomial coefficients n+1 choose 2 or n+2 choose 3, which are always integers, and then show
that it is divisible by 6.
Ideals
The set of x such that d divides x is something called an ideal of the integers. An ideal is just
something that's closed under addition and subtraction.
Euclid's Division Algorithm
The algorithm is the number of times you have to take the remainder to subtract a, denoted by r,
will be the number to subtract and the remainder r. We also have a quotient. We repeat this
process until there are no more copies of a left to subtract from b. If a is negative, you wouldn't
add copies, but instead subtract one. If b is negative, you would have to add copies of b or a to
subtract a from b. If both a and b are positive, you would subtract a from b.
The Greatest Common Divisor and Euclid's Algorithm
The greatest common divisor (GCD) is the largest integer that divides two numbers without
leaving a remainder. We can use Euclid's division algorithm to find the GCD:
Suppose we have two numbers, a and b.
If a or b is zero, the GCD is the other number.
Otherwise, we find the largest integer d such that d divides both a and b.
However, finding the GCD of large numbers can be difficult. Euclid's algorithm is a more efficient
method:
Factor a and b into primes.
Find the product of all common prime factors, raised to the minimum power.
The result is the GCD.
While Euclid's algorithm is still difficult for extremely large numbers, it is widely used in modern
computing. For example, it is used to solve equations and to factor large numbers.
The Greatest Common Divisor
