A prime number is a whole number greater than $1$ with exactly two factors: $1$ and itself.

$2$ is the only even prime — every other even number can be divided by $2$, so it automatically has more than two factors. $1$ is not prime, since it only has one factor.

A factor of a number divides into it exactly, with no remainder. The factors of $12$ are $1, 2, 3, 4, 6, 12$.

A multiple of a number is what you get by multiplying it by a whole number. The multiples of $6$ are $6, 12, 18, 24, \dots$

Use a factor tree: repeatedly split the number into two factors until every branch ends in a prime.

Example: write $210$ as a product of primes.

1. Split: $210 = 2 \times 105$. $2$ is prime — circle it.
2. Split $105 = 3 \times 35$. $3$ is prime — circle it.
3. Split $35 = 5 \times 7$. Both are prime — circle both.
4. Collect every circled number:

If a question asks for the answer in index form, group repeated factors using powers, e.g. $180 = 2^2 \times 3^2 \times 5$.

Once two numbers are written as products of primes, you don't need to list every factor by hand:

• HCF (highest common factor): multiply the lowest power of each prime that appears in both numbers.
• LCM (lowest common multiple): multiply the highest power of each prime that appears in either number.

Example: $A = 2^2 \times 3 \times 7$ and $B = 2 \times 3^2 \times 5$.

For the HCF, only $2$ and $3$ are shared by both, so take the smaller power of each:

For the LCM, take the highest power of every prime present anywhere:

A rational number can be written as a fraction $\frac{a}{b}$ with integers $a, b$ (and $b \neq 0$). This includes all integers, all terminating decimals, and — importantly — all recurring decimals like $0.\overline{3}$, since $0.\overline{3} = \frac{1}{3}$.

An irrational number cannot be written as a fraction. Its decimal expansion goes on forever with no repeating pattern, e.g. $\pi = 3.14159\dots$ and $\sqrt{2} = 1.41421\dots$

The reciprocal of a number $x$ is $\dfrac{1}{x}$. Multiplying any number by its reciprocal gives $1$.

Given that $A = 2^3 \times 3 \times 5^2$ and $B = 2^2 \times 3^3 \times 5$, find (a) the HCF and (b) the LCM of $A$ and $B$.

(a) Take the smallest power of each prime common to both:

(b) Take the largest power of every prime present:

• Calling $1$ a prime number. A prime needs exactly two factors — $1$ only has one (itself), so it's excluded by definition.
• Assuming all recurring decimals are irrational. A decimal that repeats forever in a pattern is still rational, because it converts to a fraction. Only non-repeating decimals are irrational.
• Writing an expanded product when "index form" is requested. If the question says "in index form", write $2^2 \times 3^2 \times 5$, not $2 \times 2 \times 3 \times 3 \times 5$ — you lose the accuracy mark otherwise.