Are you allowed to use the prime decomposition of integers?GlassBones said:Homework Statement
Let ##n## be odd and a composite number, prove that all of its prime is at most ##\frac{n}{3} ##
Homework Equations
Some theorems might help?
Any ##n>1## must have a prime factor
if n is composite then there is a prime ##p<√n## such that ##p|n##
The Attempt at a Solution
I'm at a standstill, not sure how to start off this question? Can anyone prove hints to get me going
The fundamental theorem of arithmetic says, that every integer can be written as a product of primes. If you are allowed to use this, and proofs can easily be found, e.g. on Wikipedia, then you only have to write this in a formula and think about what the smallest prime in there is.GlassBones said:I'm pretty sure I can (prof didn't talk about it but I'm assuming we should know about it).
With prime decomposition, I'm not sure how to generate it. Can we just assume there is a prime decomposition because n is composite?
Prime factors are the numbers that, when multiplied together, result in a given number. They are the building blocks of composite numbers.
Odd composites are numbers that are greater than 1 and have more than two factors. They are not prime numbers, and their prime factors are all odd.
To find the prime factors of an odd composite number, you can use a process called prime factorization. This involves dividing the number by its smallest prime factor, then dividing the result by its smallest prime factor, and so on until all factors are prime.
Knowing the prime factors of odd composites can be useful in many mathematical applications, such as finding the greatest common factor or simplifying fractions. It can also help in cryptography and number theory problems.
The largest prime factor of an odd composite number is always smaller than the square root of the number. This is because if a number has a prime factor larger than its square root, then it would also have a smaller prime factor, making it a composite number.