Sec 1 Maths Primes, HCF and LCM worksheet with answers

A free Sec 1 Maths worksheet on primes, hcf and lcm: 6 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 Maths worksheet: Primes, HCF and LCM

Total: 30 marksSuggested time: 45 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)It is given that

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

. Without using a calculator, find

\sqrt[3]{74088}

Answer (a)[1]

(b)It is given that

3600 = 2^{4} \times 3^{2} \times 5^{2}

. Using this, explain whether

3600

is a perfect square. Choose the answer with its reason.

Answer (b)[2]

Nur Aisyah and a friend start jogging round a track from the same point at the same time. Nur Aisyah takes

10

minutes for each round and the friend takes

15

minutes. Find the number of minutes before they are first at the starting point together again.

Answer[3]

(a)It is given that

120 = 2^{3} \times 3 \times 5

and

315 = 3^{2} \times 5 \times 7

. Find the smallest positive whole number

n

such that

120n

is a multiple of

315

Answer (a)[3]

(b)Two numbers,

A

and

B

, are written as products of their prime factors:

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

and

B = 2^{b} \times 3^{2} \times 5 \times 7

, where

a

and

b

are positive whole numbers. The HCF of

A

and

B

2 \times 3^{2}

and their LCM is

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

. Find the value of

a

and the value of

b

Answer (b)[2]

Section B

(a)It is given that

132 = 2^{2} \times 3 \times 11

and

100 = 2^{2} \times 5^{2}

. Find the smallest positive whole number

n

such that

132n

is a multiple of

100

Answer (a)[2]

(b)(i)It is given that

64000 = 2^{9} \times 5^{3}

. Without using a calculator, find

\sqrt[3]{64000}

Answer (b)(i)[1]

(b)(ii)Find the smallest positive whole number

k

such that

20k

is a perfect square.

Answer (b)(ii)[2]

(a)Find the smallest positive whole number

k

such that

120k

is a perfect square.

Answer (a)[2]

(b)A cuboid has a volume of

690

^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.

Answer (b)cm[2]

(c)Find the highest common factor (HCF) of

24

60

and

84

Answer (c)[3]

(a)(i)State whether

97

is a prime number.

Answer (a)(i)[2]

(a)(ii)Express

4056

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

120

and

144

Answer (b)(i)[3]

(b)(ii)It is given that

1200 = 2^{4} \times 3 \times 5^{2}

and

1500 = 2^{2} \times 3 \times 5^{3}

. Find the highest common factor (HCF) of

1200

and

1500

, giving your answer in index notation.

Answer (b)(ii)[1]

END OF PAPER

Answers and worked solutions

Question 1

(a)Answer:

= 42

[1]

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

= 42

(b)Answer: YES[2]

Index of

2

4

is even.Index of

3

2

is even.Index of

5

2

is even.Yes,

3600

is a perfect square: every index is even.

Question 2

Answer:

= 30

[3]

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

10 = 2 \times 5

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

15 = 3 \times 5

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

= 2 \times 3 \times 5

= 30

Question 3

(a)Answer: Check:

120n = 2^{3} \times 3^{2} \times 5 \times 7

, a multiple of

315

.[3]

3

315

has

3^{2}

120

has

3^{1}

5

315

has

5^{1}

120

has

5^{1}

: enough

7

315

has

7^{1}

120

has none

n = 3 \times 7

= 21

Check:

120n = 2^{3} \times 3^{2} \times 5 \times 7

, a multiple of

315

(b)Answer:

b = 1

[2]

Prime

3

: the LCM has

3^{3}

, the HIGHER power; B has

3^{2}

, so the higher one is A's.

a = 3

Prime

2

: the HCF has

2^{1}

, the LOWER power; A has

2^{2}

, so the lower one is B's.

b = 1

Question 4

(a)Answer: Check:

132n = 2^{2} \times 3 \times 5^{2} \times 11

, a multiple of

100

.[2]

2

100

has

2^{2}

132

has

2^{2}

: enough

5

100

has

5^{2}

132

has none

n = 5^{2}

= 25

Check:

132n = 2^{2} \times 3 \times 5^{2} \times 11

, a multiple of

100

(b)(i)Answer:

= 40

[1]

\sqrt[3]{64000} = 2^{3} \times 5

= 40

(b)(ii)Answer: Check:

20k = 2^{2} \times 5^{2}

, and every power is even.[2]

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

20 = 2^{2} \times 5

k = 5

Check:

20k = 2^{2} \times 5^{2}

, and every power is even.

Question 5

(a)Answer: Check:

120k = 2^{4} \times 3^{2} \times 5^{2}

, and every power is even.[2]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;120\;} \\ 2 & \underline{\;60\;} \\ 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

120 = 2^{3} \times 3 \times 5

k = 2 \times 3 \times 5

= 30

Check:

120k = 2^{4} \times 3^{2} \times 5^{2}

, and every power is even.

(b)Answer: The cuboid measures

5

cm by

6

cm by

23

cm (the only way with every edge greater than

3

cm and all different).[2]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;690\;} \\ 3 & \underline{\;345\;} \\ 5 & \underline{\;115\;} \\ 23 & \underline{\;23\;} \\ & \;1 \end{array}

690 = 2 \times 3 \times 5 \times 23

690 = 5 \times 6 \times 23

The cuboid measures

5

cm by

6

cm by

23

cm (the only way with every edge greater than

3

cm and all different).

(c)Answer:

= 12

[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{\;60\;} \\ 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;84\;} \\ 2 & \underline{\;42\;} \\ 3 & \underline{\;21\;} \\ 7 & \underline{\;7\;} \\ & \;1 \end{array}

24 = 2^{3} \times 3

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

84 = 2^{2} \times 3 \times 7

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

= 2^{2} \times 3

= 12

Question 6

(a)(i)Answer: YES[2]

\sqrt{97} = 9.84\ldots

Primes to test:

2, 3, 5, 7

None of

2, 3, 5, 7

divides

97

exactly.So

97

is a prime number.

(a)(ii)Answer:

4056 = 2^{3} \times 3 \times 13^{2}

[1]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;4056\;} \\ 2 & \underline{\;2028\;} \\ 2 & \underline{\;1014\;} \\ 3 & \underline{\;507\;} \\ 13 & \underline{\;169\;} \\ 13 & \underline{\;13\;} \\ & \;1 \end{array}

4056 = 2^{3} \times 3 \times 13^{2}

(b)(i)Answer:

= 24

[3]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;120\;} \\ 2 & \underline{\;60\;} \\ 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

120 = 2^{3} \times 3 \times 5

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;144\;} \\ 2 & \underline{\;72\;} \\ 2 & \underline{\;36\;} \\ 2 & \underline{\;18\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

144 = 2^{4} \times 3^{2}

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

= 2^{3} \times 3

= 24

(b)(ii)Answer:

2^2*3*5^2

[1]

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

Practice questions by skill

Sec 1 Maths Primes, HCF and LCM worksheet with answers

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

Need the formulas first? Read the Primes, HCF and LCM notes. Or practise it online with step-by-step help when you get one wrong.

Section A

(a)It is given that

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

. Without using a calculator, find

\sqrt[3]{74088}

Answer (a)[1]

(b)It is given that

3600 = 2^{4} \times 3^{2} \times 5^{2}

. Using this, explain whether

3600

is a perfect square. Choose the answer with its reason.

Answer (b)[2]

Section B

(a)It is given that

132 = 2^{2} \times 3 \times 11

and

100 = 2^{2} \times 5^{2}

. Find the smallest positive whole number

n

such that

132n

is a multiple of

100

Answer (a)[2]

(b)(i)It is given that

64000 = 2^{9} \times 5^{3}

. Without using a calculator, find

\sqrt[3]{64000}

Answer (b)(i)[1]

(b)(ii)Find the smallest positive whole number

k

such that

20k

is a perfect square.

Answer (b)(ii)[2]

Answers and worked solutions

Question 1

(a)Answer:

= 42

[1]

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

= 42

(b)Answer: YES[2]

Index of

2

4

is even.Index of

3

2

is even.Index of

5

2

is even.Yes,

3600

is a perfect square: every index is even.

Question 2

Answer:

= 30

[3]

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

10 = 2 \times 5

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

15 = 3 \times 5

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

= 2 \times 3 \times 5

= 30

Question 3

(a)Answer: Check:

120n = 2^{3} \times 3^{2} \times 5 \times 7

, a multiple of

315

.[3]

3

315

has

3^{2}

120

has

3^{1}

5

315

has

5^{1}

120

has

5^{1}

: enough

7

315

has

7^{1}

120

has none

n = 3 \times 7

= 21

Check:

120n = 2^{3} \times 3^{2} \times 5 \times 7

, a multiple of

315

(b)Answer:

b = 1

[2]

Prime

3

: the LCM has

3^{3}

, the HIGHER power; B has

3^{2}

, so the higher one is A's.

a = 3

Prime

2

: the HCF has

2^{1}

, the LOWER power; A has

2^{2}

, so the lower one is B's.

b = 1

Question 4

(a)Answer: Check:

132n = 2^{2} \times 3 \times 5^{2} \times 11

, a multiple of

100

.[2]

2

100

has

2^{2}

132

has

2^{2}

: enough

5

100

has

5^{2}

132

has none

n = 5^{2}

= 25

Check:

132n = 2^{2} \times 3 \times 5^{2} \times 11

, a multiple of

100

(b)(i)Answer:

= 40

[1]

\sqrt[3]{64000} = 2^{3} \times 5

= 40

(b)(ii)Answer: Check:

20k = 2^{2} \times 5^{2}

, and every power is even.[2]

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

20 = 2^{2} \times 5

k = 5

Check:

20k = 2^{2} \times 5^{2}

, and every power is even.

Question 5

(a)Answer: Check:

120k = 2^{4} \times 3^{2} \times 5^{2}

, and every power is even.[2]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;120\;} \\ 2 & \underline{\;60\;} \\ 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

120 = 2^{3} \times 3 \times 5

k = 2 \times 3 \times 5

= 30

Check:

120k = 2^{4} \times 3^{2} \times 5^{2}

, and every power is even.

(b)Answer: The cuboid measures

5

cm by

6

cm by

23

cm (the only way with every edge greater than

3

cm and all different).[2]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;690\;} \\ 3 & \underline{\;345\;} \\ 5 & \underline{\;115\;} \\ 23 & \underline{\;23\;} \\ & \;1 \end{array}

690 = 2 \times 3 \times 5 \times 23

690 = 5 \times 6 \times 23

The cuboid measures

5

cm by

6

cm by

23

cm (the only way with every edge greater than

3

cm and all different).

(c)Answer:

= 12

[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{\;60\;} \\ 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;84\;} \\ 2 & \underline{\;42\;} \\ 3 & \underline{\;21\;} \\ 7 & \underline{\;7\;} \\ & \;1 \end{array}

24 = 2^{3} \times 3

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

84 = 2^{2} \times 3 \times 7

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

= 2^{2} \times 3

= 12

Question 6

(a)(i)Answer: YES[2]

\sqrt{97} = 9.84\ldots

Primes to test:

2, 3, 5, 7

None of

2, 3, 5, 7

divides

97

exactly.So

97

is a prime number.

(a)(ii)Answer:

4056 = 2^{3} \times 3 \times 13^{2}

[1]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;4056\;} \\ 2 & \underline{\;2028\;} \\ 2 & \underline{\;1014\;} \\ 3 & \underline{\;507\;} \\ 13 & \underline{\;169\;} \\ 13 & \underline{\;13\;} \\ & \;1 \end{array}

4056 = 2^{3} \times 3 \times 13^{2}

(b)(i)Answer:

= 24

[3]

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;120\;} \\ 2 & \underline{\;60\;} \\ 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}

120 = 2^{3} \times 3 \times 5

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;144\;} \\ 2 & \underline{\;72\;} \\ 2 & \underline{\;36\;} \\ 2 & \underline{\;18\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

144 = 2^{4} \times 3^{2}

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

= 2^{3} \times 3

= 24

(b)(ii)Answer:

2^2*3*5^2

[1]

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