Section 1: Real numbers

The HCF of two positive integers is $18$ and their product is $6480$. What is their LCM?

Which of the following is the prime factorisation of $3960$?

Let $a$ $= 2^3 \times 3^2 \times 5$ and $b$ $= 2^2 \times 3^3 \times 7$. What is the LCM of $a$ and $b$?

What is the largest number that divides $398$ and $436$ leaving remainders $7$ and $11$ respectively?

Three bells in a temple ring every $16$, $20$ and $24$ minutes. They ring together at $10{:}30$ am. When will they next ring together?

A hall floor measures $10$ m $56$ cm by $6$ m $72$ cm. It is to be covered with identical square tiles, as large as possible, with no tile cut. How many tiles are needed?

$12^3 \times 18^2$ is written as $2^m \times 3^n$. What is the value of $m$?

The HCF of $2^3 \times 3^m \times 5$ and $2^n \times 3^4 \times 7$ is $108$, where $m$ and $n$ are positive integers. What is the LCM of the two numbers?

Two two-digit numbers have HCF $15$ and LCM $450$. What are the two numbers?

The LCM of two numbers is $14$ times their HCF, and the sum of the HCF and the LCM is $600$. If one of the numbers is $280$, what is the other?

What is the smallest perfect square that is divisible by each of $6$, $8$ and $15$?

What is the smallest number which leaves remainder $7$ when divided by each of $12$, $18$ and $30$?

What is the smallest positive integer $k$ for which $\sqrt{540k}$ is a whole number?

Which of the following numbers is irrational?

To prove that $\sqrt{3}$ is irrational, suppose $\sqrt{3}$ $= \frac{p}{q}$, where $p$ and $q$ are positive integers with no common factor. Then $p^2$ $= 3q^2$, so $3$ divides $p$. Writing $p$ $= 3m$, which equation follows?

To prove that $3 + 2\sqrt{5}$ is irrational, a student supposes that $3 + 2\sqrt{5}$ $= r$, where $r$ is rational. Which expression for $\sqrt{5}$ should come next, to show that $\sqrt{5}$ would then be rational?

Which of the following numbers is rational?

How many of these seven numbers are irrational? $\sqrt{2.25}$, $(1 + \sqrt{2})^2$, $\sqrt{8} \times \sqrt{2}$, $\frac{\sqrt{50}}{\sqrt{2}}$, $(\sqrt{3} - 1)(\sqrt{3} + 1)$, $\sqrt{12} + \sqrt{3}$, $2\pi$

$a$ is a positive integer and $7$ divides $a^2$. Which statement must be true?

$a$ is a rational number and $(2\sqrt{3} + a)(\sqrt{3} - 1)$ is rational. What is the value of $(2\sqrt{3} + a)(\sqrt{3} - 1)$?

In the proof that $\sqrt{5}$ is irrational, the step from $p^2$ $= 5q^2$ to the statement that $5$ divides $p$ uses which fact?

Let $x$ $= 4 - \sqrt{7}$. For which value of $y$ are both $x + y$ and $xy$ rational?

Assertion (A): $6 + \sqrt{2}$ is an irrational number. Reason (R): The sum of a rational number and an irrational number is irrational.

Assertion (A): $\sqrt{7} \times \sqrt{28}$ is a rational number. Reason (R): The product of two irrational numbers is always irrational.

Assertion (A): $15^n$ ends with the digit $0$ for some natural number $n$. Reason (R): A natural number ends with the digit $0$ only if its prime factorisation contains both $2$ and $5$.

What is the greatest number of boxes the organisers can make?

When the sweets are packed into the greatest possible number of identical boxes, how many sweets are in each box?

For the next fete, the stall wants equal numbers of laddoos and barfis, bought in whole trays, using as few trays as possible. How many trays does it buy in all?