Sec 1 G2 Maths Primes, HCF and LCM worksheet with answers

A free Sec 1 G2 Maths worksheet on primes, hcf and lcm: 5 questions, 30 marks, short questions first and then structured ones, in the school exam style. The answers and full worked solutions are on the last page.

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Sec 1 G2 Maths worksheet: Primes, HCF and LCM

Total: 30 marksSuggested time: 50 minutes

Name: ______________________________Class: ____________ Date: ____________

Answer all questions. Show all your working. Give non-exact answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.

Section A

(a)Two bus services leave an interchange together at 7.00 a.m. One leaves every

18

minutes and the other every

24

minutes. Find the number of minutes before they next leave together.

Answer (a)[3]

(b)A cuboid has a volume of

660

^3

. Its length, breadth and height are whole numbers of centimetres, all different, and each is greater than

5

cm. Find its length, breadth and height.

Answer (b)cm[2]

(a)Find the lowest common multiple (LCM) of

45

and

60

Answer (a)[3]

(b)Two numbers,

A

and

B

, are written as products of their prime factors:

A = 2 \times 3 \times 5^{a}

and

B = 3^{b} \times 5

, where

a

and

b

are positive whole numbers. The HCF of

A

and

B

3 \times 5

and their LCM is

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

. Find the value of

a

and the value of

b

Answer (b)[2]

Section B

(a)Find the smallest positive whole number

k

such that

108k

is a perfect cube.

Answer (a)[2]

(b)A box in the shape of a cuboid measures

24

cm by

36

cm by

60

cm. It is completely filled with identical cubes, each as large as possible, with no gaps. Find the number of cubes in the box.

Answer (b)[3]

(a)(i)It is given that

35937 = 3^{3} \times 11^{3}

. Without using a calculator, find

\sqrt[3]{35937}

Answer (a)(i)[1]

(a)(ii)It is given that

74088 = 2^{3} \times 3^{3} \times 7^{3}

. Without using a calculator, find

\sqrt[3]{74088}

Answer (a)(ii)[1]

(b)Ravi says, "

119

is a prime number." Is Ravi correct? Choose the answer with its reason.

Answer (b)[2]

(c)A florist has

18

roses and

27

lilies. She uses all of them to make identical bouquets, each with the same number of roses and the same number of lilies. Find the largest number of bouquets she can make.

Answer (c)[3]

(a)(i)Express

1176

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

Answer (a)(i)[1]

(a)(ii)Express

2205

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

Answer (a)(ii)[1]

(b)(i)Find the highest common factor (HCF) of

24

and

40

Answer (b)(i)[3]

(b)(ii)Find the highest common factor (HCF) of

12

18

and

42

Answer (b)(ii)[3]

END OF PAPER

Answers and worked solutions

Question 1

(a)Answer:

= 72

[3]

\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}

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;24\;} \\ 2 & \underline{\;12\;} \\ 2 & \underline{\;6\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

24 = 2^{3} \times 3

The smallest number that every amount divides into: the LCM.LCM

= 2^{3} \times 3^{2}

= 72

(b)Answer: The cuboid measures

6

cm by

10

cm by

11

cm (the only way with every edge greater than

5

cm and all different).[2]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;660\;} \\ 2 & \underline{\;330\;} \\ 3 & \underline{\;165\;} \\ 5 & \underline{\;55\;} \\ 11 & \underline{\;11\;} \\ & \;1 \end{array}

660 = 2^{2} \times 3 \times 5 \times 11

660 = 6 \times 10 \times 11

The cuboid measures

6

cm by

10

cm by

11

cm (the only way with every edge greater than

5

cm and all different).

Question 2

(a)Answer:

= 180

[3]

\def\arraystretch{1.8}\begin{array}{r|l} 3 & \underline{\;45\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

45 = 3^{2} \times 5

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;60\;} \\ 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

60 = 2^{2} \times 3 \times 5

Take every prime that appears, each with its highest power.LCM

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

= 180

(b)Answer:

b = 3

[2]

Prime

5

: the LCM has

5^{2}

, the HIGHER power; B has

5^{1}

, so the higher one is A's.

a = 2

Prime

3

: the LCM has

3^{3}

, the HIGHER power; A has

3^{1}

, so the higher one is B's.

b = 3

Question 3

(a)Answer: Check:

108k = 2^{3} \times 3^{3}

, and every power is a multiple of 3.[2]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;108\;} \\ 2 & \underline{\;54\;} \\ 3 & \underline{\;27\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

108 = 2^{2} \times 3^{3}

k = 2

Check:

108k = 2^{3} \times 3^{3}

, and every power is a multiple of 3.

(b)Answer:

= 30

[3]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;24\;} \\ 2 & \underline{\;12\;} \\ 2 & \underline{\;6\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;36\;} \\ 2 & \underline{\;18\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;60\;} \\ 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

24 = 2^{3} \times 3

36 = 2^{2} \times 3^{2}

60 = 2^{2} \times 3 \times 5

HCF

= 2^{2} \times 3

= 12

\dfrac{24}{12} \times \dfrac{36}{12} \times \dfrac{60}{12} = 2 \times 3 \times 5

= 30

Question 4

(a)(i)Answer:

= 33

[1]

\sqrt[3]{35937} = 3 \times 11

= 33

(a)(ii)Answer:

= 42

[1]

\sqrt[3]{74088} = 2 \times 3 \times 7

= 42

(b)Answer: FACTOR[2]

A prime number has exactly

2

factors:

1

and itself.

119 \div 7 = 17

119

has the factors

1

7

17

and

119

, so it is not prime: Ravi is not correct.

(c)Answer:

= 9

[3]

\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}

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

27 = 3^{3}

The largest number that divides every amount exactly: the HCF.HCF

= 3^{2}

= 9

Question 5

(a)(i)Answer:

1176 = 2^{3} \times 3 \times 7^{2}

[1]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;1176\;} \\ 2 & \underline{\;588\;} \\ 2 & \underline{\;294\;} \\ 3 & \underline{\;147\;} \\ 7 & \underline{\;49\;} \\ 7 & \underline{\;7\;} \\ & \;1 \end{array}

1176 = 2^{3} \times 3 \times 7^{2}

(a)(ii)Answer:

2205 = 3^{2} \times 5 \times 7^{2}

[1]

\def\arraystretch{1.8}\begin{array}{r|l} 3 & \underline{\;2205\;} \\ 3 & \underline{\;735\;} \\ 5 & \underline{\;245\;} \\ 7 & \underline{\;49\;} \\ 7 & \underline{\;7\;} \\ & \;1 \end{array}

2205 = 3^{2} \times 5 \times 7^{2}

(b)(i)Answer:

= 8

[3]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;24\;} \\ 2 & \underline{\;12\;} \\ 2 & \underline{\;6\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

24 = 2^{3} \times 3

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

40 = 2^{3} \times 5

Take only the primes common to every number, each with its lowest power.HCF

= 2^{3}

= 8

(b)(ii)Answer:

= 6

[3]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;12\;} \\ 2 & \underline{\;6\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

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

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;42\;} \\ 3 & \underline{\;21\;} \\ 7 & \underline{\;7\;} \\ & \;1 \end{array}

12 = 2^{2} \times 3

18 = 2 \times 3^{2}

42 = 2 \times 3 \times 7

Take only the primes common to every number, each with its lowest power.HCF

= 2 \times 3

= 6

Sec 1 G2 Maths Primes, HCF and LCM worksheet with answers

In the print window, choose Save as PDF to download it.

Or practise it online with step-by-step help when you get one wrong.

Section A

(a)Two bus services leave an interchange together at 7.00 a.m. One leaves every

18

minutes and the other every

24

minutes. Find the number of minutes before they next leave together.

Answer (a)[3]

(b)A cuboid has a volume of

660

^3

. Its length, breadth and height are whole numbers of centimetres, all different, and each is greater than

5

cm. Find its length, breadth and height.

Answer (b)cm[2]

Section B

(a)Find the smallest positive whole number

k

such that

108k

is a perfect cube.

Answer (a)[2]

(b)A box in the shape of a cuboid measures

24

cm by

36

cm by

60

cm. It is completely filled with identical cubes, each as large as possible, with no gaps. Find the number of cubes in the box.

Answer (b)[3]

Answers and worked solutions

Question 1

(a)Answer:

= 72

[3]

\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}

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;24\;} \\ 2 & \underline{\;12\;} \\ 2 & \underline{\;6\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

24 = 2^{3} \times 3

The smallest number that every amount divides into: the LCM.LCM

= 2^{3} \times 3^{2}

= 72

(b)Answer: The cuboid measures

6

cm by

10

cm by

11

cm (the only way with every edge greater than

5

cm and all different).[2]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;660\;} \\ 2 & \underline{\;330\;} \\ 3 & \underline{\;165\;} \\ 5 & \underline{\;55\;} \\ 11 & \underline{\;11\;} \\ & \;1 \end{array}

660 = 2^{2} \times 3 \times 5 \times 11

660 = 6 \times 10 \times 11

The cuboid measures

6

cm by

10

cm by

11

cm (the only way with every edge greater than

5

cm and all different).

Question 2

(a)Answer:

= 180

[3]

\def\arraystretch{1.8}\begin{array}{r|l} 3 & \underline{\;45\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

45 = 3^{2} \times 5

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;60\;} \\ 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

60 = 2^{2} \times 3 \times 5

Take every prime that appears, each with its highest power.LCM

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

= 180

(b)Answer:

b = 3

[2]

Prime

5

: the LCM has

5^{2}

, the HIGHER power; B has

5^{1}

, so the higher one is A's.

a = 2

Prime

3

: the LCM has

3^{3}

, the HIGHER power; A has

3^{1}

, so the higher one is B's.

b = 3

Question 3

(a)Answer: Check:

108k = 2^{3} \times 3^{3}

, and every power is a multiple of 3.[2]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;108\;} \\ 2 & \underline{\;54\;} \\ 3 & \underline{\;27\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

108 = 2^{2} \times 3^{3}

k = 2

Check:

108k = 2^{3} \times 3^{3}

, and every power is a multiple of 3.

(b)Answer:

= 30

[3]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;24\;} \\ 2 & \underline{\;12\;} \\ 2 & \underline{\;6\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;36\;} \\ 2 & \underline{\;18\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;60\;} \\ 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

24 = 2^{3} \times 3

36 = 2^{2} \times 3^{2}

60 = 2^{2} \times 3 \times 5

HCF

= 2^{2} \times 3

= 12

\dfrac{24}{12} \times \dfrac{36}{12} \times \dfrac{60}{12} = 2 \times 3 \times 5

= 30

Question 4

(a)(i)Answer:

= 33

[1]

\sqrt[3]{35937} = 3 \times 11

= 33

(a)(ii)Answer:

= 42

[1]

\sqrt[3]{74088} = 2 \times 3 \times 7

= 42

(b)Answer: FACTOR[2]

A prime number has exactly

2

factors:

1

and itself.

119 \div 7 = 17

119

has the factors

1

7

17

and

119

, so it is not prime: Ravi is not correct.

(c)Answer:

= 9

[3]

\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}

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

27 = 3^{3}

The largest number that divides every amount exactly: the HCF.HCF

= 3^{2}

= 9

Question 5

(a)(i)Answer:

1176 = 2^{3} \times 3 \times 7^{2}

[1]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;1176\;} \\ 2 & \underline{\;588\;} \\ 2 & \underline{\;294\;} \\ 3 & \underline{\;147\;} \\ 7 & \underline{\;49\;} \\ 7 & \underline{\;7\;} \\ & \;1 \end{array}

1176 = 2^{3} \times 3 \times 7^{2}

(a)(ii)Answer:

2205 = 3^{2} \times 5 \times 7^{2}

[1]

\def\arraystretch{1.8}\begin{array}{r|l} 3 & \underline{\;2205\;} \\ 3 & \underline{\;735\;} \\ 5 & \underline{\;245\;} \\ 7 & \underline{\;49\;} \\ 7 & \underline{\;7\;} \\ & \;1 \end{array}

2205 = 3^{2} \times 5 \times 7^{2}

(b)(i)Answer:

= 8

[3]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;24\;} \\ 2 & \underline{\;12\;} \\ 2 & \underline{\;6\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

24 = 2^{3} \times 3

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

40 = 2^{3} \times 5

Take only the primes common to every number, each with its lowest power.HCF

= 2^{3}

= 8

(b)(ii)Answer:

= 6

[3]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;12\;} \\ 2 & \underline{\;6\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

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

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;42\;} \\ 3 & \underline{\;21\;} \\ 7 & \underline{\;7\;} \\ & \;1 \end{array}

12 = 2^{2} \times 3

18 = 2 \times 3^{2}

42 = 2 \times 3 \times 7

Take only the primes common to every number, each with its lowest power.HCF

= 2 \times 3

= 6