Sec 1 · G3 (Express)

Primes, HCF and LCM

Practise primes, hcf and lcm with step-by-step help. Free account.

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}

📖 Worked example

A cuboid has a volume of $390$ cm $^3$ . Its length, breadth and height are whole numbers of centimetres, all different, and each is greater than $3$ cm. Find its length, breadth and height.

Step 1 of 2

Short division of 390 by primes, smallest first, until you reach 1; then 390 in index notation

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;390\;} \\ 3 & \underline{\;195\;} \\ 5 & \underline{\;65\;} \\ 13 & \underline{\;13\;} \\ & \;1 \end{array}

390 = 2 \times 3 \times 5 \times 13

📖 Worked example

Priya says, " $87$ is a prime number." Is Priya correct? Choose the answer with its reason.

Step 1 of 3

A prime number has exactly two different factors: 1 and itself

A prime number has exactly

2

factors:

1

and itself.

Practise this →

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}

📖 Worked example

Two ribbons are $30$ cm and $42$ cm long. Siti cuts both into pieces that are all the same length, with nothing left over. Find the greatest possible length of each piece.

Step 1 of 4

Short division of 30 by primes, smallest first, until you reach 1; then 30 in index notation

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

30 = 2 \times 3 \times 5

📖 Worked example

It is given that $120 = 2^{3} \times 3 \times 5$ and $132 = 2^{2} \times 3 \times 11$ . Find the smallest positive whole number $n$ such that $120n$ is a multiple of $132$ .

Step 1 of 2

120n is a multiple of 132 when it has every prime of 132, each to at least the same power: compare the powers prime by prime

2

132

has

2^{2}

120

has

2^{3}

: enough

3

132

has

3^{1}

120

has

3^{1}

: enough

11

132

has

11^{1}

120

has none

📖 Worked example

Two numbers, $A$ and $B$ , are written as products of their prime factors: $A = 2^{3} \times 3^{a} \times 5$ and $B = 2^{b} \times 3^{3}$ , where $a$ and $b$ are positive whole numbers. The HCF of $A$ and $B$ is $2^{2} \times 3$ and their LCM is $2^{3} \times 3^{3} \times 5$ . Find the value of $a$ and the value of $b$ .

Step 1 of 1

HCF: the LOWER power of each common prime. LCM: the HIGHER power of every prime. Compare the powers of each prime with an unknown

Prime

3

: the HCF has

3^{1}

, the LOWER power; B has

3^{3}

, so the lower one is A's.

a = 1

Prime

2

: the HCF has

2^{2}

, the LOWER power; A has

2^{3}

, so the lower one is B's.

b = 2

📖 Worked example

Two whole numbers have a highest common factor of $2$ and a lowest common multiple of $28$ . Neither number is equal to $2$ . Find the two numbers.

Step 1 of 3

Each number is the HCF times a factor; the two factors have no common factor and multiply to LCM ÷ HCF

\dfrac{28}{2} = 14

14 = 2 \times 7

📖 Worked example

A box in the shape of a cuboid measures $30$ cm by $45$ cm by $75$ cm. It is completely filled with identical cubes, each as large as possible, with no gaps. Find the number of cubes in the box.

Step 1 of 6

Short division of 30 by primes, smallest first, until you reach 1

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

📖 Worked example

Two bus services leave an interchange together at 0710. Service A leaves every $30$ minutes and service B leaves every $45$ minutes. Find the time at which they next leave the interchange together. Give your answer using the 24-hour clock.

Step 1 of 3

They are together again after a time that every interval divides into: the LCM

30 = 2 \times 3 \times 5

45 = 3^{2} \times 5

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

\phantom{\text{LCM}} = 90

minutes

Practise this →

3. Square roots and cube roots

Square root: halve each index

\displaystyle \sqrt{2^{4} \times 3^{2}}

e.g.

\displaystyle \sqrt{2^{4} \times 3^{2}}

\displaystyle = 2^{2} \times 3

\displaystyle = 12

Cube root: divide each index by 3

\displaystyle \sqrt[3]{2^{6} \times 5^{3}}

e.g.

\displaystyle \sqrt[3]{2^{6} \times 5^{3}}

\displaystyle = 2^{2} \times 5

\displaystyle = 20

Number	Index rule	Example
perfect square	every index is even	$144 = 2^{4} \times 3^{2}$
perfect cube	every index is a multiple of $3$	$8000 = 2^{6} \times 5^{3}$

📖 Worked example

Express $676$ as a product of its prime factors. Hence find $\sqrt{676}$ without using a calculator.

Step 1 of 3

Short division of 676 by primes, smallest first, until you reach 1

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;676\;} \\ 2 & \underline{\;338\;} \\ 13 & \underline{\;169\;} \\ 13 & \underline{\;13\;} \\ & \;1 \end{array}

📖 Worked example

It is given that $288 = 2^{5} \times 3^{2}$ . Find the smallest positive whole number $k$ such that $\dfrac{288}{k}$ is a perfect square.

Step 1 of 1

Dividing takes powers away: remove just enough of each prime to leave an EVEN power

k = 2

= 2

Check:

\dfrac{288}{k} = 2^{4} \times 3^{2}

, and every power is even.

📖 Worked example

It is given that $2000 = 2^{4} \times 5^{3}$ . Using this, explain whether $2000$ is a perfect square. Choose the answer with its reason.

Step 1 of 2

A perfect square has every index EVEN (its square root halves them)

Index of

2

4

is even.Index of

5

3

is not even.

Practise this →

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Sec 1 · G3 (Express)

Primes, HCF and LCM

Practise primes, hcf and lcm with step-by-step help. Free account.

Start free Worksheet (PDF)

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}

📖 Worked example

A cuboid has a volume of $390$ cm $^3$ . Its length, breadth and height are whole numbers of centimetres, all different, and each is greater than $3$ cm. Find its length, breadth and height.

Step 1 of 2

Short division of 390 by primes, smallest first, until you reach 1; then 390 in index notation

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;390\;} \\ 3 & \underline{\;195\;} \\ 5 & \underline{\;65\;} \\ 13 & \underline{\;13\;} \\ & \;1 \end{array}

390 = 2 \times 3 \times 5 \times 13

📖 Worked example

Priya says, " $87$ is a prime number." Is Priya correct? Choose the answer with its reason.

Step 1 of 3

A prime number has exactly two different factors: 1 and itself

A prime number has exactly

2

factors:

1

and itself.

Practise this →

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}

📖 Worked example

Two ribbons are $30$ cm and $42$ cm long. Siti cuts both into pieces that are all the same length, with nothing left over. Find the greatest possible length of each piece.

Step 1 of 4

Short division of 30 by primes, smallest first, until you reach 1; then 30 in index notation

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

30 = 2 \times 3 \times 5

📖 Worked example

It is given that $120 = 2^{3} \times 3 \times 5$ and $132 = 2^{2} \times 3 \times 11$ . Find the smallest positive whole number $n$ such that $120n$ is a multiple of $132$ .

Step 1 of 2

120n is a multiple of 132 when it has every prime of 132, each to at least the same power: compare the powers prime by prime

2

132

has

2^{2}

120

has

2^{3}

: enough

3

132

has

3^{1}

120

has

3^{1}

: enough

11

132

has

11^{1}

120

has none

📖 Worked example

Step 1 of 1

HCF: the LOWER power of each common prime. LCM: the HIGHER power of every prime. Compare the powers of each prime with an unknown

Prime

3

: the HCF has

3^{1}

, the LOWER power; B has

3^{3}

, so the lower one is A's.

a = 1

Prime

2

: the HCF has

2^{2}

, the LOWER power; A has

2^{3}

, so the lower one is B's.

b = 2

📖 Worked example

Two whole numbers have a highest common factor of $2$ and a lowest common multiple of $28$ . Neither number is equal to $2$ . Find the two numbers.

Step 1 of 3

Each number is the HCF times a factor; the two factors have no common factor and multiply to LCM ÷ HCF

\dfrac{28}{2} = 14

14 = 2 \times 7

📖 Worked example

A box in the shape of a cuboid measures $30$ cm by $45$ cm by $75$ cm. It is completely filled with identical cubes, each as large as possible, with no gaps. Find the number of cubes in the box.

Step 1 of 6

Short division of 30 by primes, smallest first, until you reach 1

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

📖 Worked example

Step 1 of 3

They are together again after a time that every interval divides into: the LCM

30 = 2 \times 3 \times 5

45 = 3^{2} \times 5

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

\phantom{\text{LCM}} = 90

minutes

Practise this →

3. Square roots and cube roots

Square root: halve each index

\displaystyle \sqrt{2^{4} \times 3^{2}}

e.g.

\displaystyle \sqrt{2^{4} \times 3^{2}}

\displaystyle = 2^{2} \times 3

\displaystyle = 12

Cube root: divide each index by 3

\displaystyle \sqrt[3]{2^{6} \times 5^{3}}

e.g.

\displaystyle \sqrt[3]{2^{6} \times 5^{3}}

\displaystyle = 2^{2} \times 5

\displaystyle = 20

Number	Index rule	Example
perfect square	every index is even	$144 = 2^{4} \times 3^{2}$
perfect cube	every index is a multiple of $3$	$8000 = 2^{6} \times 5^{3}$

📖 Worked example

Express $676$ as a product of its prime factors. Hence find $\sqrt{676}$ without using a calculator.

Step 1 of 3

Short division of 676 by primes, smallest first, until you reach 1

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;676\;} \\ 2 & \underline{\;338\;} \\ 13 & \underline{\;169\;} \\ 13 & \underline{\;13\;} \\ & \;1 \end{array}

📖 Worked example

It is given that $288 = 2^{5} \times 3^{2}$ . Find the smallest positive whole number $k$ such that $\dfrac{288}{k}$ is a perfect square.

Step 1 of 1

Dividing takes powers away: remove just enough of each prime to leave an EVEN power

k = 2

= 2

Check:

\dfrac{288}{k} = 2^{4} \times 3^{2}

, and every power is even.

📖 Worked example

It is given that $2000 = 2^{4} \times 5^{3}$ . Using this, explain whether $2000$ is a perfect square. Choose the answer with its reason.

Step 1 of 2

A perfect square has every index EVEN (its square root halves them)

Index of

2

4

is even.Index of

5

3

is not even.

Practise this →

Try a question: Primes, HCF and LCM

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise primes, hcf and lcm with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free