1.1 Types of Number

Quick Notes & Formulae

Essential concepts for solving types of number problems.

Prime Factorization

Break down numbers into products of prime factors. Use factor trees or repeated division.

H C F

= product of lowest powers of common primes.

L C M

= product of highest powers of all primes.

Rational vs Irrational

Rational: Can be written as

\frac{a}{b}

where

a, b

are integers (

b \neq 0

). Includes terminating and recurring decimals. Irrational: Non-terminating, non-recurring decimals like

π

\sqrt{2}

Number Sets Hierarchy

N \subset Z \subset Q \subset R

. Natural ⊂ Integers ⊂ Rational ⊂ Real. Irrationals are also in Real but not in Rational.

Divisibility Rules

By 2: last digit even. By 3: sum of digits divisible by 3. By 4: last two digits divisible by 4. By 5: last digit 0 or 5.

Number of Factors

n = p^{a} \times q^{b} \times r^{c}

, then number of factors =

(a + 1) (b + 1) (c + 1)

. For exactly 9 factors:

n = p^{8}

n = p^{2} q^{2}

Past Paper Style Questions

Question 1: Prime Factors → HCF/LCM

(a) Write 84 as a product of its prime factors. [2]

(b) Write 126 as a product of its prime factors. [2]

(d) Hence find the LCM of 84 and 126. [2]

(e) Explain briefly why your HCF and LCM answers are consistent with the prime factor forms. [2]

Question 2: Reverse Question - Build Numbers from HCF/LCM

Two positive integers are written as:

A = 2^{3} \times 3^{m}, B = 2^{n} \times 3^{2}

and

HCF (A, B) = 2^{2} \times 3^{2}, LCM (A, B) = 2^{5} \times 3^{m}

(a) Find the values of

m

and

n

. [3]

(b) Work out the values of

A

and

B

. [2]

A

is a square number, giving a reason. [2]

(d) Find the number of positive factors of

B

. [2]

Question 3: Rational / Irrational and Exact Forms

(a) State whether each number is rational or irrational. Give a reason. [4]

(i)

\sqrt{49}

(ii)

\sqrt{50}

(iii)

0.125

(iv)

0. \overset{―}{36}

(recurring)

(b) Write

0. \overset{―}{36}

as a fraction in its simplest form. [3]

\sqrt{50} - \sqrt{2}

is rational or irrational. Explain. [2]

(d) Write

\sqrt{50}

in the form

k \sqrt{2}

, where

k

is an integer. [1]

Question 4: Ordering Mixed Number Types

(a) Arrange the following in ascending order: [3]

- 2.7, - \frac{8}{3}, \sqrt{7}, 2.6, π

(b) Write down the smallest and the largest number from the list. [1]

(d) Give one reason why

\sqrt{7}

is between 2 and 3. [1]

Question 5: Squares, Cubes, and 'Between' Reasoning

A positive integer

N

satisfies:

$N$ is a multiple of 6
$N$ is between 70 and 110
$N$ is one less than a square number

(a) Find

N

. [3]

(b) Write the square number as

k^{2}

and state

k

. [1]

N

as a product of prime factors. [2]

(d) How many positive factors does

N

have? [2]

Question 6: Place Value, Rounding, and Standard Form

A population is recorded as

5 083 496

(a) Write this number in words. [1]

(b) State the value of the digit 8. [1]

(d) Round the number to 3 significant figures. [1]

(e) Write the original number in standard form. [2]

Question 7: Reverse Algebraic Digits Question

A two-digit number is

10 a + b

, where

a

and

b

are integers and

1 \leq a \leq 9

0 \leq b \leq 9

The number is divisible by 3 and by 4, and the sum of its digits is 12.

(a) Form an equation in

a

and

b

using the digit sum condition. [1]

(b) Use the divisibility rule for 4 to write a condition involving

a

and

b

. [2]

(d) For each number found, state whether it is prime or composite. [2]

(e) Classify one of your numbers as natural, integer, rational, irrational, or real (choose all that apply). [2]

Question 8: Common Multiples in Context

A bus leaves a station every 18 minutes. A train leaves the same station every 30 minutes. At 09:00, both leave together.

(a) Find the next time after 09:00 when they both leave together. [3]

(b) How many times will they leave together between 09:00 and 12:00 inclusive? [2]

(d) If the train schedule changes to every 24 minutes, repeat part (a). [3]

Question 9: Real Numbers and Number Sets

(a) Put each number into the most suitable set(s): Natural numbers, Integers, Rational numbers, Irrational numbers, Real numbers. [5]

- 7, 0, \frac{11}{4}, \sqrt{81}, \sqrt{10}, 0.2020020002 . . .

(b) Explain why

0.2020020002 . . .

is not rational. [1]

\sqrt{81}

as an integer and state whether it is a square number. [2]

(d) Give an example of a rational number between

\sqrt{10}

and 4. [1]

Question 10: Reverse Factors Question

A positive integer

n

is less than 100 and has exactly 9 positive factors.

(a) Explain why

n

must be of the form

p^{8}

p^{2} q^{2}

, where

p

and

q

are primes. [3]

(b) Find all possible values of

n

less than 100. [3]

n

, write it as a product of prime factors. [2]

(d) Choose one value of

n

and list all of its factors in ascending order. [2]

Question 11: Advanced HCF/LCM with Three Numbers

Given that

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

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

, and

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

(a) Find the HCF of A, B, and C. [2]

(b) Find the LCM of A, B, and C. [2]

\frac{A \times B}{HCF (A, B)}

. [3]

(d) Find the smallest positive integer

k

such that

A \times k

is a perfect square. [3]

Question 12: Simplifying Surds and Exact Values

(a) Simplify

\sqrt{72} + \sqrt{50} - \sqrt{8}

. Give your answer in the form

a \sqrt{b}

where

a

and

b

are integers and

b

is as small as possible. [4]

(b) Show that

(3 \sqrt{2} + 2 \sqrt{3})^{2} = 30 + 12 \sqrt{6}

. [3]

\frac{5}{\sqrt{7} - 2}

. [3]

(d) Given that

\sqrt{45} + \sqrt{20} = k \sqrt{5}

, find the value of

k

. [2]

Question 13: Recurring Decimals and Fractions

(a) Express

0. \overset{―}{7}

as a fraction in its simplest form. [2]

(b) Express

0.2 \overset{―}{45}

(i.e., 0.2454545...) as a fraction in its simplest form. [3]

0. \overset{―}{9} = 1

. [2]

(d) The recurring decimal

0. \overset{―}{a b}

where

a

and

b

are digits, equals

\frac{7}{33}

. Find the digits

a

and

b

. [3]

Question 14: Prime Numbers and Goldbach's Conjecture

(a) List all prime numbers between 40 and 60. [2]

(b) Express 50 as the sum of two prime numbers in two different ways. [2]

p > 20

such that

p + 2

is also prime (twin primes). [2]

(d) A student claims: "All prime numbers except 2 are odd, so the sum of any two primes (excluding 2) is always even." Is the student correct? Explain. [2]

Question 15: Bounds and Limits of Accuracy

A rectangular field has length

l = 85

m (to the nearest metre) and width

w = 42

m (to the nearest metre).

(a) Write down the lower and upper bounds for

l

and

w

. [2]

(b) Calculate the lower and upper bounds for the area of the field. [3]

(c) The field is to be fenced at a cost of $25 per metre (to the nearest dollar). Find the maximum possible cost of fencing the perimeter. [3]

(d) Given that

x = 4.5

(to 1 decimal place) and

y = 3.25

(to 2 decimal places), calculate the lower bound of

\frac{x}{y}

. [2]

Question 16: Indices and Standard Form

(a) Evaluate

\frac{2^{- 3} \times 3^{4}}{2^{- 5} \times 3^{2}}

, leaving your answer as a simplified fraction. [3]

(b) Simplify

(\frac{8 x^{6}}{27 y^{9}})^{\frac{1}{3}}

. [3]

p = 4.5 \times 10^{5}

and

q = 1.5 \times 10^{- 2}

, calculate:

(i)

p \times q

in standard form [2]

(ii)

\frac{p}{q}

in standard form [2]

(d) Light travels at approximately

3 \times 10^{8}

m/s. The distance from Earth to the Sun is approximately

1.5 \times 10^{11}

m. Calculate how long it takes light to travel from the Sun to Earth. Give your answer in minutes. [3]

Question 17: Estimation and Approximation

(a) Estimate the value of

\frac{49.7 \times 10.2}{0.49}

. [2]

(b) By writing each number correct to 1 significant figure, estimate

\sqrt{\frac{82.3}{0.198}}

. [3]

\sqrt{150} = 12.24744871 . . .

(i) Round this to 3 significant figures. [1]

(ii) Round this to 3 decimal places. [1]

(d) Show that

\sqrt{150}

is irrational. [2]

Question 18: Number Patterns and Sequences

(a) The

n

th term of a sequence is given by

T_{n} = n^{2} + n + 1

(i) Find the first three terms of the sequence. [2]

(ii) Show that all terms are odd numbers. [2]

(b) The sum of the first

n

odd numbers is

n^{2}

. Verify this for

n = 5

. [2]

(i) Find the next term. [1]

(ii) Find the

n

th term formula. [2]

Question 19: Egyptian Fractions and Unit Fractions

Ancient Egyptians expressed fractions as sums of unit fractions (fractions with numerator 1).

(a) Express

\frac{2}{3}

as the sum of two different unit fractions. [2]

(b) Express

\frac{3}{4}

as the sum of three different unit fractions. [2]

\frac{1}{n} - \frac{1}{n + 1} = \frac{1}{n (n + 1)}

. [2]

(d) Use the result from part (c) to express

\frac{1}{6}

as the difference of two unit fractions. [2]

Question 20: Cryptarithmetic and Digit Puzzles

In this question, different letters represent different digits.

(a) Find digits A and B such that

\overset{―}{A B} + \overset{―}{B A} = 121

, where

\overset{―}{A B}

represents the two-digit number with digits A and B. [3]

(b) Solve for digits C and D:

\overset{―}{C D} \times 4 = \overset{―}{D C}

. [3]

\overset{―}{x y z}

, show that

37 x = 2 y + 5 z

. [3]

(d) Find one three-digit number that satisfies the condition in part (c). [1]