Every number belongs to one or more of these sets:

A prime number has exactly two factors, $1$ and itself. A reciprocal of $x$ is $\dfrac{1}{x}$ ($x \neq 0$).

Repeatedly divide by the smallest prime that fits, until only primes remain, then collect the results using index notation. Every integer greater than $1$ has exactly one such breakdown — the Fundamental Theorem of Arithmetic.

Write $A$ and $B$ in prime-factor form, then:

and always $\text{HCF}(A,B) \times \text{LCM}(A,B) = A \times B$ — a handy check.

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

A number is a perfect square exactly when every exponent in its prime factorisation is even, and a perfect cube exactly when every exponent is a multiple of $3$. To turn a number into a perfect square (or cube) with the smallest possible multiplier, factorise it and top up whichever exponents are "short".

Worked Example

Find the smallest positive integer $k$ such that $288k$ is a perfect square.

1. Factorise: $288 = 2^5 \times 3^2$.
2. The exponent of $2$ is $5$ (odd) — one more factor of $2$ makes it $6$ (even). The exponent of $3$ is already even.
3. So $k = 2$, and $288 \times 2 = 576 = 24^2$. ✓

Ask whether it can be written exactly as a fraction $\dfrac{p}{q}$ of two integers. A decimal that terminates (like $0.75$) or recurs (like $0.\dot{3}$) is always rational, because both types convert to an exact fraction. A decimal that goes on forever with no repeating pattern — such as $\pi$ or $0.101001000100001\ldots$ — is irrational.

Roots are the most common trap: $\sqrt{n}$ is irrational unless $n$ is a perfect square, in which case the root symbol is hiding a whole number in disguise.