## Prime Real Estate

One of the most important concepts to learn before taking a standardised exam is how to think about primes – unfortunately, it’s precisely this part of number theory that most students never get a firm grounding in! Getting to grips with primes allows you to find Least Common Multiples and Greatest Common Factors, add and subtract fractions with ease, and even take the roots of non-perfect-squares in your head!

To see why primes are so important, consider this: primes are the only integers divisible by only two different factors – themselves and 1. Which means every other integer can be broken down into factors that are either prime, or not prime – and the not-prime factors can be broken down the same way. What this means is that every integer in this world can be broken down into its prime factors – and that’s what the number really is! 18 is really 2 times 9, which itself is 3 times 3; so really, 18 is just one 2 and two 3s multiplied together! 18 = $2^1 3^2$

Here’s where the fun starts. If we want to take the square root of 18, all we have to do it look for pairs of prime factors – in this case, the two 3s. We can take the square root of $3^2$, since $3^2$ is 9 – and $\sqrt{9}$ is 3! We can’t take the root of 2, though – so we leave it alone. Consequently, the square root of 18 is simply $3\sqrt{2}$

Try it yourself! Break down the following numbers and see what roots you can take.

• 8
• 27
• 50
• 98
• 54
• 252
• 3,000,000,000,000 (for funzies…)