Sec 1 · G1 (N(T))

Primes, HCF and LCM

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1. Prime numbers and prime factorisation

Term	Meaning	Example
prime number	exactly two factors: $1$ and itself	$2, 3, 5, 7, 11, 13$
composite number	more than two factors	$4, 6, 8, 9, 10$
neither	$1$ has only one factor	$1$
prime factorisation	the number written as a product of its prime factors	$360 = 2^{3} \times 3^{2} \times 5$ (index notation)

Divide by primes until you reach 1

\displaystyle 360 = 2^{3} \times 3^{2} \times 5

Repeated division of 360: by 2 to 180, by 2 to 90, by 2 to 45, by 3 to 15, by 3 to 5, by 5 to 1.

the primes down the left side are the prime factors

📖 Worked example

Express $40$ as a product of its prime factors, giving your answer in index notation.

Step 1 of 2

Short division: divide by the smallest prime that goes in exactly, again and again, until you reach 1

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;40\;} \\ 2 & \underline{\;20\;} \\ 2 & \underline{\;10\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

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2. HCF and LCM

	$2$	$3$	$5$	$7$
$360$	$2^{3}$	$3^{2}$	$5$
$84$	$2^{2}$	$3$		$7$
HCF: lowest power of each common prime	$2^{2}$	$3$
LCM: highest power of every prime	$2^{3}$	$3^{2}$	$5$	$7$

HCF of 360 and 84

\displaystyle \text{HCF} = 2^{2} \times 3

e.g.

\displaystyle 2^{2} \times 3

\displaystyle = 12

LCM of 360 and 84

\displaystyle \text{LCM} = 2^{3} \times 3^{2} \times 5 \times 7

e.g.

\displaystyle 2^{3} \times 3^{2} \times 5 \times 7

\displaystyle = 2520

Word problem asks for	Use
the largest equal group, the longest equal piece	HCF
the next time things happen together, the smallest common amount	LCM

📖 Worked example

Find the highest common factor (HCF) of $27$ and $45$ .

Step 1 of 4

Short division of 27 by primes, smallest first, until you reach 1; then 27 in index notation

\def\arraystretch{1.8}\begin{array}{r|l} 3 & \underline{\;27\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

27 = 3^{3}

📖 Worked example

Find the lowest common multiple (LCM) of $18$ and $30$ .

Step 1 of 4

Short division of 18 by primes, smallest first, until you reach 1; then 18 in index notation

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;18\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

18 = 2 \times 3^{2}

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1. Prime numbers and prime factorisation

Term	Meaning	Example
prime number	exactly two factors: $1$ and itself	$2, 3, 5, 7, 11, 13$
composite number	more than two factors	$4, 6, 8, 9, 10$
neither	$1$ has only one factor	$1$
prime factorisation	the number written as a product of its prime factors	$360 = 2^{3} \times 3^{2} \times 5$ (index notation)

Divide by primes until you reach 1

\displaystyle 360 = 2^{3} \times 3^{2} \times 5

the primes down the left side are the prime factors

📖 Worked example

Express $40$ as a product of its prime factors, giving your answer in index notation.

Step 1 of 2

Short division: divide by the smallest prime that goes in exactly, again and again, until you reach 1

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;40\;} \\ 2 & \underline{\;20\;} \\ 2 & \underline{\;10\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

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2. HCF and LCM

	$2$	$3$	$5$	$7$
$360$	$2^{3}$	$3^{2}$	$5$
$84$	$2^{2}$	$3$		$7$
HCF: lowest power of each common prime	$2^{2}$	$3$
LCM: highest power of every prime	$2^{3}$	$3^{2}$	$5$	$7$

HCF of 360 and 84

\displaystyle \text{HCF} = 2^{2} \times 3

e.g.

\displaystyle 2^{2} \times 3

\displaystyle = 12

LCM of 360 and 84

\displaystyle \text{LCM} = 2^{3} \times 3^{2} \times 5 \times 7

e.g.

\displaystyle 2^{3} \times 3^{2} \times 5 \times 7

\displaystyle = 2520

Word problem asks for	Use
the largest equal group, the longest equal piece	HCF
the next time things happen together, the smallest common amount	LCM

📖 Worked example

Find the highest common factor (HCF) of $27$ and $45$ .

Step 1 of 4

Short division of 27 by primes, smallest first, until you reach 1; then 27 in index notation

\def\arraystretch{1.8}\begin{array}{r|l} 3 & \underline{\;27\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

27 = 3^{3}

📖 Worked example

Find the lowest common multiple (LCM) of $18$ and $30$ .

Step 1 of 4

Short division of 18 by primes, smallest first, until you reach 1; then 18 in index notation

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;18\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

18 = 2 \times 3^{2}

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