Number theory is the branch of mathematics that studies integers, which are all the whole numbers on either side of the number line. Number theory looks at specific properties of integers and seeks patterns in the ways different types of numbers are distributed or related to each other.
The following are a few of the topics a course on number theory would likely address, along with a few examples of each.
1. Divisibility rules
Divisibility rules are tools to help you know quickly whether a number is divisible by a certain integer. The following are a few sample rules.
- All even numbers (ending in 0, 2, 4, 6, or 8) are divisible by 2. For instance, 1,104 is divisible by 2 because its last digit, 4, is divisible by 2.
- A number is divisible by 3 if the sum of its digits is divisible by three. For example, the number 288 is divisible by 3 because 2+8+8=18, which is divisible by 3.
- A number is divisible by 6 if it’s divisible by both 2 and 3. In the second example above, we established that 288 is divisible by 3. Because it ends in an even number, it’s also divisible by 2, meaning that 288 is divisible by 6.
2. Factors
Factors are two whole numbers that, when multiplied together, equal a third number. All numbers except 0 and 1 have at least two factors: 1 and the number itself. But numbers may have many more factors. The number 100, for example, has 9 factors: 1, 2, 4, 5, 10, 20, 25, 50, and 100.
3. Prime numbers
Prime numbers are a special set of numbers that have only 2 distinct factors: 1 and the number itself. The number 11 is prime, for example, because its only factors are 1 and 11. The number 12, on the other hand, is a composite (non-prime) number, because it has 5 different factors: 1, 2, 3, 4, 6, and 12.
Mathematicians are interested in prime numbers because they represent the building blocks of all the numbers that exist. This means that every composite number can be represented as the product of prime factors. For example, 100 = 2 x 2 x 5 x 5. Primes are also very interesting because there is still a lot that is not yet know about them.
Number Theory Problems And Solutions
Many basic number theory problems relate to factoring. Following are a couple of examples:
Example 1
Problem: You have a quantity of cookies. You can share them among 2 people or 3 people or 4 people equally. What is the minimum number of cookies you can have to fulfill these conditions?
Solution: The answer is 12 because 2, 3, and 4 are all factors of 12, and 12 is the lowest common multiple of those numbers.
Example 2
Problem: Which of the following numbers can not be divided into any smaller equal groups: 5106, 5281, or 5751?
Solution: 5281 is a prime number, so it cannot be subdivided into smaller equal groups. It can be found through process of elimination. 5106 ends in an even number, so it must be divisible by 2. In the case of 5751, the sum of its digits (5+7+5+1=18) is divisible by three, so 5751 must be divisible by 3.
Applications Of Number Theory
One of the most well-known applications of number theory is cryptography, particularly online. Modern cryptography depends on prime factorization of extremely large numbers. Number theory has also contributed greatly to the development of computer science.