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anemone
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For positive integers $p,\,q,\,r,\,s$ such that $ps=q^2+qr+r^2$, prove that $p^2+q^2+r^2+s^2$ is a composite number.
That possibility is easily eliminated. If $p-q-r+s = 1$ then the equation becomes $p^2+q^2+r^2+s^2 = p+q+r+s$. That can only hold for positive integers if $p=q=r=s=1$. But in that case the equation $ps = q^2+qr+r^2$ does not hold.kaliprasad said:if one talks of algebraic expression being composite I agree but if one talks of the number being composite this is not correct as one of the factors could be 1
Opalg said:That possibility is easily eliminated. If $p-q-r+s = 1$ then the equation becomes $p^2+q^2+r^2+s^2 = p+q+r+s$. That can only hold for positive integers if $p=q=r=s=1$. But in that case the equation $ps = q^2+qr+r^2$ does not hold.
Opalg said:That possibility is easily eliminated. If $p-q-r+s = 1$ then the equation becomes $p^2+q^2+r^2+s^2 = p+q+r+s$. That can only hold for positive integers if $p=q=r=s=1$. But in that case the equation $ps = q^2+qr+r^2$ does not hold.
A composite number is a positive integer that has more than two factors. In other words, it can be divided evenly by numbers other than 1 and itself. Examples of composite numbers include 4, 6, 8, and 10.
To prove that a sum is a composite number, you need to show that it has more than two factors. This can be done by finding two or more numbers that evenly divide the sum. If the sum can be divided by at least three numbers (not including 1 and itself), then it is a composite number.
No, a sum of two prime numbers will always be a prime number itself. This is because prime numbers only have two factors (1 and itself), so adding two prime numbers will not result in any additional factors.
An example of a sum that is a composite number is 12. It can be divided evenly by 2, 3, 4, and 6, in addition to 1 and itself. Therefore, it has more than two factors and is considered a composite number.
There are several techniques for determining if a number is prime or composite, but there is no universal shortcut or formula for proving a sum is a composite number. The best approach is to try dividing the sum by different numbers and see if there are more than two factors. If so, the sum is composite.