A number when divided by 53 gives 34 as quotient and 21 as remainder. Find the number.

4

(a)

The product of two consecutive positive integers is divisible by 2. Is this statement true or false? Give reason.

4

(a)

Use the Euclid's division algorithm to find the HCF of
(i) 650 and 1170
(ii) 870 and 225

4

(a)

If two positive integers p and q can be expressed as p = ab² and q = a³b; where a,b being prime numbers, find the LCM (p,q)

4

(a)

Find the HCF and LCM of 60, 84 and 108 by using the prime factorisation method.

4

(a)

Show that \(\left( \sqrt { 3 } +\sqrt { 5 } \right) ^{ 2 }\) is an irrational number.

4

(a)

Use Euclid's division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1, for some integer m.

4

(a)

Prove that \(3+2\sqrt { 5 } \) is irrational.

4

(a)

Write the missing numbers is the following factorisation.
(i)       (ii) 

4

(a)

Prove that n²-n is divisible by 2 for every positive integer n.

4

(a)