Types of number：练习题

Which of the following numbers is an irrational number?

Identify the rational number from the list below.

Simplify the expression involving surds:
\( \sqrt{48} - \sqrt{12} \)

Find the Highest Common Factor (HCF) of \( 36 \) and \( 54 \).

Two buses, Route X and Route Y, leave the central station at the same time. Route X buses leave every \( 15 \) minutes and Route Y buses leave every \( 25 \) minutes. After how many minutes will both buses next leave the station at the same time?

List all the prime numbers that are between \( 10 \) and \( 20 \).

State whether \( \sqrt{8} \) is rational or irrational and give a brief reason for your answer.

Given the set of numbers \( S = \{ -4, \sqrt{2}, 0, \frac{22}{7}, \pi, 9 \} \):
(a) List all the integers in the set.
(b) List all the rational numbers in the set.

From the following list of numbers:
\( 4, 9, 15, 21, 25, 27 \)

(a) Find the highest common factor (HCF) of \( 24 \) and \( 60 \).
(b) From the list provided, identify all numbers that are square numbers and also odd numbers.

Triangle numbers follow the pattern 1, 3, 6, 10, ... where the \(n\)-th term is given by the formula \(T_n = \frac{n(n+1)}{2}\).

(a) Show that the 5th triangle number is 15.
(b) Find the 10th triangle number.
(c) Determine if the sum of the 3rd and 4th triangle numbers is a square number. Show your working.