When written as a product of their prime factors, $a=n\times 3\times 7$ and $b=n^2\times 7$.
(a) Express, in terms of $n$:
(i) The LCM of $a$ and $b$,

(ii) The HCF of $a$ and $b$.

(b) Given that the LCM of $a$ and $b$ is 84.
Find the values of $n$, $a$ and $b$.

Given that $180=5\times p^2\times q^2$ and that both $p$ and $q$ are both prime.
Find the values of $p$ and $q$.

Here is a list of numbers:
36, 29, 41, 45, 15, 10, 13
(a) Find the median,

(b) Find the probability that a number chosen at random from this list is prime.

(c) Thirteen more numbers are added to the list.
The smallest number is 14 while the largest number is 48.
(i) Calculate the range of the 20 numbers now in the list.

(ii) The mean of all the twenty numbers is 19.2.
Calculate the mean of the added thirteen numbers.

Here is a list of numbers:
7, 5, 13, 5, 13, 4, 11, $x$, $y$
The mean of the numbers is 8 and the mode is 5.
(a) Find the value of $x$ and the value of $y$, where $x<y$,

(b) The median,

(c) The probability that two numbers selected at random without replacement are both five.

(a) Express 520 as a product of its primes.

(b) It is given that $1260=2^2\times 3^2\times 5\times 7$.
Find the simplest integer value of $k$ such that $(1260\times 520k)$ is a square number.

Find the LCM of 702 and 1690.
Leave your answer as a product of prime factors in index form.

(a) Express 120 as a product of its prime factors.

(b) Given that 120 $k$ is a perfect square, write down the smallest possible value of $k$.

(c) Find the highest common factor of 120 and 180.

21, 22, 23, 24, 25, 26, 27, 28, 29
From the list of numbers write:
(a) A prime number,

(b) A square number,

(c) A cube number,

(d) The square root of 625.

$198=2\times 3^2\times 11$ and $360=2^3\times 3^2\times 5$
(a) Find:
(i) The highest common factor of 198 and 360,

(ii) The lowest common multiple of 198 and 360.

(b) Find the smallest value of $k$ such that $\sqrt{360\times 198\times k}$ is an integer.

Natural numbers P, Q and R, are such that: Find:
(i) P and Q

(ii) The smallest value R for which $\sqrt{369R}$ is a natural number.

(a) Write 126 as a product of its prime factors.

(b) $A=3^2\times 5^{q+3}\times p^3$ and $B=3^3\times 5^3\times p^2$ where p is prime and q is a positive integer.
The highest common factor of A and B is 135 000.
Find the value of:
(i) p,

(ii) q.

(c) Find the difference of the fifth prime number and the fourth composite number.

(d) Show that $2.\overline{12}$ is rational.

(a) Evaluate $\sqrt[3]{{(-2.01)}^2+2.8}$, giving your answer to three significant figures.

(b) $P=\frac{2}{3}m{t}^2-c$
(i) Find the value of P when c = 4, m = 2 and t = −3.

(ii) Make t the subject of the formula.