Primes, HCF and LCM - Highest Common Factor and Its Applications

Prime Numbers, Composite Numbers and Prime Factorisation (Questions 1 – 4)

A prime number is an integer greater than 1 with exactly two positive factors, 1 and itself.

A composite number is an integer greater than 1 with more than two positive factors.

• Special cases:

Zero and 1 are neither prime nor composite.

\(\circ\ 2\) is the first prime number, and the only prime number that is an even number.

• Prime factor(s) of a whole number is/are its factor(s) that is/are prime number(s).

To test whether an integer n greater than 1 is prime:

Test every prime not exceeding \(\sqrt n\). If none divides \(n\), it is prime; if any divides it, it is composite.

Prime factorisation expresses an integer greater than 1 as a product of primes, usually using index notation.

Figure 1.1 is an example of finding the prime factorisation of \(18\) using long division.

Figure 1.1 Finding the Prime Factorisation of \(18\) Using Long Division

Highest Common Factor and Lowest Common Multiple (Questions 5 – 8, 23 – 25 )

1. Methods to find the highest common factor (HCF) of two or more numbers:

\(\circ\ \) By Prime Factorisation

Find the prime factorisation of each number in index notation;

For the HCF of positive integers, take the common prime factors to their lowest powers. If no prime is common to all numbers, the HCF is 1.

\(\circ\ \) By Long Division.

Start from dividing each number by the smallest common prime factors, until there are no common prime factors for all of the numbers.

Then the HCF of these numbers is the product of the common prime factors on the left.

2. Methods to find the lowest common multiple (LCM) of two or more numbers:

\(\circ\ \) By Prime Factorisation:

Find the prime factorisation of each number in index notation;

Then the LCM of these numbers is the product of all of the prime factors that are each to the highest power in the prime factorisations.

\(\circ\ \) By Long Division.

For the LCM, repeatedly choose a prime dividing at least two current numbers. Divide the numbers it divides and carry the others unchanged, until the remaining numbers are pairwise coprime.

Then the LCM of these numbers is the product of the common prime factors on the left and all of the remaining factors.

Square Roots and Cube Roots Involving Prime Factorisation (Questions 9 – 14)

• A number that is the square of a whole number is known as a perfect square (or square number).

For \(a\ge0\), \(\sqrt a\) denotes the nonnegative principal square root. If \(a>0\), the equation \(x^2=a\) has the two solutions \(x=\pm\sqrt a\).

An integer that is the cube of an integer is a perfect cube. Negative perfect cubes are possible; this prime-factorisation section considers positive integers.

The real cube root of \(a\), denoted \(\sqrt[3]{a}\), is the unique real number whose cube is \(a\).

• Prime factorisation can be used to determine the square/cube root of a given number or determine whether a given number is a perfect square/cube or not.

Applications of HCF and LCM (Questions 15 – 22, 26)

Finding the HCF and LCM of numbers are applied under many real-world situations to evaluate the numerical property of data or provide us with quantitative reference in decision-making. For more specific real-world applications of HCF and LCM, see questions of this unit.