1.3 Index Notation and Prime Factorisation

Index notation writes repeated multiplication short. In $5^3 = 125$ , the base is $5$ and the index is $3$ .

Index notation

a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}

Prime factorisation writes a composite as a product of primes. Every composite has exactly ONE such product however you reach it, and two methods do.

Split into any two factors, then keep splitting the composite ones until every branch ends on a prime. Any first split reaches the same answer.

Factor tree for 360

\begin{array}{ccccccccc} & & & & 360 & & & & \\ & & & \swarrow & & \searrow & & & \\ & & 4 & & & & 90 & & \\ & \swarrow & & \searrow & & \swarrow & & \searrow & \\ \boxed{2} & & & \boxed{2} & 9 & & & 10 \\ & & & & \swarrow \; \searrow & & \swarrow & \searrow & \\ & & & \boxed{3} & & \boxed{3} & \boxed{2} & & \boxed{5} \end{array}

Divide by the smallest prime that goes in, repeatedly, until you reach $1$ .

Try it

A practice question on this appears here. Practise primes hcf lcm.

Repeated division (ladder method) for 360

\begin{array}{r|l} 2 & 360 \\ 2 & 180 \\ 2 & 90 \\ 3 & 45 \\ 3 & 15 \\ 5 & 5 \\ & 1 \end{array} \quad \Rightarrow \quad 360 = 2^3 \times 3^2 \times 5

Finding prime factorisation

Express $360$ as a product of its prime factors in index notation.

Repeated division:
$360 \div 2 = 180$
$180 \div 2 = 90$
$90 \div 2 = 45$
$45 \div 3 = 15$
$15 \div 3 = 5$
$5 \div 5 = 1$
$\therefore 360 = 2^3 \times 3^2 \times 5$

Try it

A practice question on this appears here. Practise primes hcf lcm.

Using prime factorisation to solve a problem

A rectangular floor is tiled with exactly $385$ identical square tiles. Both side lengths, in tiles, are greater than $1$ . Find every possible pair.

$385 = 5 \times 7 \times 11 \quad \scriptsize\textit{prime factorisation}$
Since each dimension $> 1$ , we need two factors of $385$ , both $> 1$ :
$385 = 5 \times 77 = 7 \times 55 = 11 \times 35$
$\therefore$ the possible dimensions are $5 \times 77$ , $7 \times 55$ , or $11 \times 35$ tiles.

Practise this

Questions on Primes, HCF and LCM, marked as you go, with the working shown.

Practise Primes, HCF and LCM