Prove that the sum of 6 positive integers is a composite number

In summary, a composite number is a positive integer with more than two factors. Proving that the sum of 6 positive integers is a composite number can help us better understand composite numbers and can be used in mathematical proofs. To prove this, we can use the fact that any number can be written as a sum of two or more numbers and show that the sum is divisible by a number other than 1 and itself. An example of a sum of 6 positive integers that is a composite number is 6 + 8 + 10 + 12 + 14 + 16 = 66. There is no specific formula for finding the sum of 6 positive integers that is a composite number, but we can use principles of number
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Let $a,\,b,\,c,\,d,\,e,\,f$ be positive integers and $S=a+b+c+d+e+f$. Suppose that the number $S$ divides $abc+def$ and $ab+bc+ca-de-ef-df$, prove that $S$ is composite.
 
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All the coefficients of

$\begin{align*}f(x)&=(x+a)(x+b)(x+c)-(x-d)(x-e)(x-f)\\&=Sx^2+(ab+bc+ca-de-ef-fd)x+(abc+def)\end{align*}$

are multiples of $S$. Evaluating $f$ at $d$, we get that $f(d)=(a+d)(b+d)(c+d)$ is a multiple of $S$.

So this implies that $S$ is composite, since $a+d,\,b+d,\,c+d$ are all strictly less than $S$.
 

Related to Prove that the sum of 6 positive integers is a composite number

1. What is a composite number?

A composite number is a positive integer that can be divided evenly by at least one number other than 1 and itself. In other words, it has more than two factors.

2. How can you prove that the sum of 6 positive integers is a composite number?

To prove that the sum of 6 positive integers is a composite number, we can show that it is divisible by at least one number other than 1 and itself. This can be done by finding the prime factorization of the sum and showing that it has more than two factors.

3. Can the sum of 6 positive integers be a prime number?

No, the sum of 6 positive integers cannot be a prime number because it is always divisible by at least 3, which means it has more than two factors and is therefore a composite number.

4. What is the smallest possible sum of 6 positive integers that is a composite number?

The smallest possible sum of 6 positive integers that is a composite number is 4, which can be shown by adding 1 + 1 + 1 + 1 + 1 + 1. This sum is divisible by 2 and 4, making it a composite number.

5. Can the sum of 6 positive integers be an even prime number?

No, the sum of 6 positive integers cannot be an even prime number because all even numbers are divisible by 2, meaning they have more than two factors and are therefore composite numbers.

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