Singapore-Cambridge O-Level Mathematics

Prime numbers and prime factorisation

A prime number is a whole number greater than 1 whose only factors are 1 and itself. The first few are 2, 3, 5, 7, 11, 13, 17, 19, 23. The number 2 is the only even prime, and 1 is not prime because it has only one factor. A whole number greater than 1 that is not prime is composite.

Every composite number can be written as a product of primes in exactly one way (apart from the order), and this is its prime factorisation. To find it, divide repeatedly by the smallest prime that goes in, and stop when the quotient is 1.

So $360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^3 \times 3^2 \times 5$. A factor tree gives the same result: split the number into any two factors, then keep splitting until every branch ends in a prime.

To test whether a number $n$ is prime, it is enough to try the primes up to $\sqrt{n}$, because any factor pair has one member no larger than $\sqrt{n}$. For 97 the primes to try are 2, 3, 5 and 7, none of which divides it, so 97 is prime.

Highest common factor

The highest common factor (HCF) of two or more numbers is the largest whole number that divides exactly into all of them. Writing each number as a product of primes makes it easy to find: take each prime that appears in every number, raised to the lowest power it has in any of them.

For $84 = 2^2 \times 3 \times 7$ and $90 = 2 \times 3^2 \times 5$, the primes common to both are 2 and 3. The lowest power of 2 is $2^1$ and the lowest power of 3 is $3^1$, so

Lowest common multiple

The lowest common multiple (LCM) of two or more numbers is the smallest whole number that each of them divides into exactly. Take every prime that appears in any of the numbers, raised to the highest power it has in any of them.

For the same two numbers, the primes involved are 2, 3, 5 and 7, with highest powers $2^2$, $3^2$, $5$ and $7$:

For exactly two numbers, $\text{HCF} \times \text{LCM}$ equals the product of the numbers. Here $6 \times 1\,260 = 7\,560 = 84 \times 90$, which is a useful check. The rule does not extend to three numbers.

The HCF answers questions about splitting into equal groups with nothing left over, such as cutting two lengths of 84 cm and 90 cm into equal pieces as long as possible (6 cm). The LCM answers questions about events that repeat together. Two buses leave a station every 12 minutes and every 18 minutes, and $12 = 2^2 \times 3$ and $18 = 2 \times 3^2$ give an LCM of $2^2 \times 3^2 = 36$, so they leave together every 36 minutes.

Squares, cubes and their roots

A number is a perfect square when every prime in its factorisation has an even power, and a perfect cube when every power is a multiple of 3. Square roots and cube roots of such numbers come from halving or thirding the powers.

The same idea finds the smallest number to multiply by to make a square or a cube. Since $252 = 2^2 \times 3^2 \times 7$, only the power of 7 is odd, so multiplying by 7 gives $2^2 \times 3^2 \times 7^2 = 1\,764$, a perfect square. For $180 = 2^2 \times 3^2 \times 5$ to become a cube, each power must reach 3, which needs one more 2, one more 3 and two more 5s. The multiplier is $2 \times 3 \times 5^2 = 150$, and $180 \times 150 = 27\,000 = 30^3$.

Finally, the squares up to $15^2 = 225$ and cubes up to $10^3 = 1\,000$ come up so often that recognising them saves time: for example $11^2 = 121$, $12^2 = 144$, $13^2 = 169$, $14^2 = 196$, $2^3 = 8$, $3^3 = 27$, $4^3 = 64$, $5^3 = 125$.