HallsofIvy said:If n is not divisible by 5, then it is of the form n= 5k+ 1, n= 5k+ 2, n= 5k+ 3, or n= 5k+ 4.
Or you can consider n= 5k- 2, n= 5k- 1, n= 5k+ 1, and n= 5k+ 2 if that makes the calculations simpler.
Divisibility is the property of a number being evenly divisible by another number without leaving a remainder. In other words, when one number is divided by another, the result is a whole number.
There are a few different methods for proving divisibility depending on the specific situation. Some common techniques include using the division algorithm, testing for divisibility by specific numbers (such as 2, 3, 5, etc.), and using mathematical induction.
The division algorithm is a mathematical procedure used to divide two numbers and obtain a quotient and remainder. It is often used to prove divisibility, as the remainder can reveal whether or not the two numbers are evenly divisible.
Mathematical induction is a technique used to prove statements about natural numbers. To use it to prove divisibility, you would first show that the statement is true for a specific number (usually 1). Then, you would assume it is true for a particular number (k), and use that assumption to prove it is also true for the next number (k+1). If you can show that the statement is true for both 1 and k+1, then it must be true for all natural numbers.
Proving divisibility is important in mathematics because it allows us to understand the relationships between numbers and identify patterns. It also helps us to solve more complex problems and make generalizations about numbers. Additionally, divisibility is a fundamental concept in number theory, which has many applications in fields such as cryptography and computer science.