Types of Number (Copy)
Practice Questions – 1.1 Types of Number
Question 1
Write the following numbers in words:
(a) 7 000 000
(b) 4 205 018
(c) 10 000 007
Question 2
Write these numbers in figures:
(a) Nine million, seventy-five thousand, two hundred and four
(b) Twelve billion, four million, thirty-five
(c) Eight hundred and ninety thousand and seven
Question 3
Identify which of the following are natural numbers, which are integers but not natural numbers, and which are not integers:
−5, 0, 7, 2.3, 100, −47.
Question 4
List all the prime numbers between 20 and 40.
Question 5
Express 180 as a product of its prime factors in index form.
Question 6
Find the highest common factor (HCF) of 84 and 126.
Question 7
Find the lowest common multiple (LCM) of 12 and 18.
Question 8
Write down the first 5 square numbers and the first 5 cube numbers.
Question 9
Determine whether each of the following numbers is rational or irrational:
(a) 0.25
(b) 3/7
(c) √5
(d) π
(e) 0.333…
Question 10
Write down the reciprocals of:
(a) 5
(b) −2
(c) 1/4
(d) 0.1
Question 11
Find the prime factorisation of 72 and 98. Use it to calculate:
(a) HCF(72, 98)
(b) LCM(72, 98).
Question 12
True or false? Give reasons.
(a) Every integer is a natural number.
(b) Every prime number is odd.
(c) Every square number has an odd number of factors.
Question 13
Simplify the following:
(a) (HCF of 96 and 60) ÷ (LCM of 6 and 8)
(b) (LCM of 12 and 20) − (HCF of 45 and 60).
Question 14
Ali says: “√81 is irrational because it has a square root.”
Explain why Ali is wrong.
Question 15
Between which two consecutive integers does √50 lie?
Answer key and explanations — 1.1 Types of number
1.
- (a) 7 000 000 = seven million
- (b) 4 205 018 = four million, two hundred and five thousand, eighteen
- (c) 10 000 007 = ten million and seven
Explanation: break the number into groups of thousands and write them in words.
2.
- (a) Nine million, seventy-five thousand, two hundred and four = 9 075 204
- (b) Twelve billion, four million, thirty-five = 12 000 004 035
- (c) Eight hundred and ninety thousand and seven = 890 007
Explanation: place-value system ensures missing groups are filled with zeros.
3.
- −5: integer but not natural
- 0: integer but not natural (since natural numbers start from 1)
- 7: natural number
- 2.3: not an integer (rational but not integer)
- 100: natural number
- −47: integer but not natural
Explanation: integers include negatives, zero, and positives. Naturals are only positive counting numbers.
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4.
Prime numbers between 20 and 40: 23, 29, 31, 37.
Explanation: check divisibility by 2, 3, or 5 only, since √40 < 7.
5.
180 = 2² × 3² × 5
Explanation: repeatedly divide by smallest prime factors until only 1 remains.
6.
HCF(84, 126) = 42
Working:
84 = 2² × 3 × 7
126 = 2 × 3² × 7
Take smallest powers of each prime → 2¹ × 3¹ × 7 = 42.
7.
LCM(12, 18) = 36
Working:
12 = 2² × 3
18 = 2 × 3²
Take highest powers → 2² × 3² = 36.
8.
First 5 square numbers: 1, 4, 9, 16, 25
First 5 cube numbers: 1, 8, 27, 64, 125
Explanation: squares are n², cubes are n³.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
9.
- 0.25 → rational (1/4)
- 3/7 → rational
- √5 → irrational
- π → irrational
- 0.333… recurring → rational (1/3)
Explanation: rational numbers can be expressed as a fraction or terminating/recurring decimal.
10.
- Reciprocal of 5 = 1/5
- Reciprocal of −2 = −1/2
- Reciprocal of 1/4 = 4
- Reciprocal of 0.1 = 10
Explanation: reciprocal is 1 ÷ number.
11.
72 = 2³ × 3²
98 = 2 × 7²
- HCF(72, 98) = 2
- LCM(72, 98) = 2³ × 3² × 7² = 3528
Explanation: HCF uses lowest prime powers, LCM uses highest.
12.
(a) False — integers can be negative or zero, not all are natural
(b) False — 2 is a prime and even
(c) True — square numbers have an odd number of factors
Explanation: factor pairs collapse into a repeated factor at the square root.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
13.
(a) HCF(96, 60) ÷ LCM(6, 8) = 12 ÷ 24 = 1/2
(b) LCM(12, 20) − HCF(45, 60) = 60 − 15 = 45
Explanation: factorize to find HCF and LCM, then substitute.
14.
Ali is wrong. √81 = 9 which is rational (an integer).
Explanation: irrational roots only occur for non-square integers.
15.
√50 lies between 7² = 49 and 8² = 64, so between 7 and 8 (approx 7.07).
Explanation: compare with nearest perfect squares.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course