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symplectic_manifold
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Prove that n^3-n is divisible by 6 for every integer n. Is it induction to be used here?...
symplectic_manifold said:If 6 divides n^3-n then, since divisibility is transitive and 2 divides 6 , 2 must also divide n^3-n. Let n be even then n^3 is even and n^3-n is also even. Let n be odd then n^3 is odd and hence n^3-n is even. Since every even number is divisible by 2 it follows that 6 divides n^3-n.
n3-n= n(n2-1)= n(n-1)(n+1)= (n-1)(n)(n+1), three consecutive integers. What does that tell you?
You've managed to show 2 divides n^3-n by considering n even or odd, but to show 6 divides n^3-n you need to also show 3 divides n^3-n. Consider Hall's factorization, can you show 3 divides (n-1)n(n+1) for any n?
symplectic_manifold said:Great!...Can I generalise it by saying that among any n successive integers there is exactly one divisible by n?...What is a formal proof for this?
symplectic_manifold said:OK, but why did you change r?
A number is divisible by 6 if it can be divided evenly by 6 without leaving a remainder. In other words, the result of the division is a whole number.
This can be proved using mathematical induction. We first show that the statement is true for n=1, and then assume that it is also true for n=k. Using this assumption, we can prove that it is also true for n=k+1. This establishes the truth of the statement for all positive integers.
Proving this statement for every integer n provides a general proof for a specific pattern. It ensures that the statement holds true for all possible values of n, rather than just a few specific cases. This can be useful in solving other mathematical problems and in making accurate predictions.
This proof has practical applications in various fields such as computer science, physics, and engineering. It can be used to analyze and solve problems involving sequences, series, and patterns. In computer science, it can be used to optimize algorithms and in physics, it can be used to understand and predict physical phenomena.
Yes, there are other methods to prove this statement such as using the Binomial Theorem or using the properties of even and odd numbers. However, using mathematical induction is the most common and efficient method to prove statements about sequences and patterns.