A positive integer is prime if it’s divisible by only 1 and itself. For example, 2, 3, 5 and 7 are prime, but 4, 6, 8 and 9 are not. The number 1, by definition, is not prime. a) Write a method that determines whether a number is prime. b) Use this method in an application that determines and displays all the prime numbers less than 10,000. How many numbers up to 10,000 do you have to test to ensure that you’ve found all the primes? c) Initially, you might think that n/2 is the upper limit for which you must test to see whether a number n is prime, but you need only go as high as the square root of n. Rewrite the program, and run it both ways.

Java is a widely used programming language known for its portability, performance, and strong object-oriented features. It operates on the principle of 'Write Once, Run Anywhere', making Java code compilable and executable on any platform that has a Java Virtual Machine (JVM).

Java's syntax is similar to C++, but with a simpler object model and fewer low-level facilities. Java also eliminates the possibility of memory corruption by managing memory through an automatic garbage collector. When solving problems like determining prime numbers, Java provides a robust platform with powerful libraries like Math, which can be incredibly useful for mathematical operations and performance optimizations.

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. When it comes to algorithms for identifying prime numbers, the basic approach is to iterate through all possible divisors and check for divisibility.

In Java, for example, you could implement a isPrime method that would use a loop to check every number from 2 to the input number minus one. If any number divides evenly into the input number (meaning the remainder is zero when using the modulus operator), the number is not prime. This algorithm, while straightforward, is not the most efficient way of finding prime numbers, especially as the input number grows larger.

The performance of prime checking can be greatly improved by implementing Math.sqrt optimization. Since any non-prime number must have a divisor no larger than its square root, it's unnecessary to check for divisors beyond the square root of the number being tested.

This significantly reduces the number of iterations required, as you only iterate up to the square root of the number, which can be obtained by calling Math.sqrt(n) in Java. Here's a simple enhancement to the prime number checking loop: instead of iterating from 2 to ‘n-1’, iterate from 2 to (int) Math.sqrt(n) and perform the same divisibility check. This improved method will have the same results but with a considerable reduction in computation time, especially for larger numbers.