Just like atoms are the building blocks of molecules, prime numbers are the building blocks of integers, or whole numbers. What do we mean by this?
Well, every molecule is composed of atoms, and each atom is an atom of a particular element, like carbon or nitrogen. Similarly, every positive integer (except \(1\)) can be broken down into prime numbers. We can say this formally as follows:
The Fundamental Theorem of Arithmetic says that any natural number greater than \(1\) can be uniquely expressed as a product of prime numbers in non-decreasing order. For example: \[630=2\times3\times3\times5\times7=2\times3^2\times5\times7.\]
Recall that a number is composite if it is a product of two smaller, natural numbers. For example, \(6 = 2\times3\). Otherwise, and if the number is not equal to \(1\), it is called prime. The number \(1\) is neither prime nor composite.
Modulo operation: We look at division. For example \(41=2\times15+11\) is the division of \(41\) (the dividend) by \(15\) (the divisor) with remainder \(11\), and \(5=0\times7+5\) is the division of \(5\) by \(7\) with the remainder \(5\). More generally, when we divide \(a\) by \(b\), we’re looking for non-negative integers \(c\) and \(d\) (\(d<b\)) such that \(a=c\times b+d\).
In the case of \(45\) divided by \(15\), we get \(3\) with remainder \(0\) - in which case we say “\(15\) divides \(45\)", or “\(45\) is divisible by \(15\)". We can write this as \(15|45\).
We can deduce from the Fundamental Theorem of Arithmetic that if a product of two natural numbers is divisible by a prime number, then one of these numbers is divisible by this prime number. For example, \(7|7007=13\times539\) tells us that \(7\) divides \(13\) or \(539\). Clearly \(7\nmid13\), so we know \(7|539\).