Number Systems

1

Given that √3 is an irrational number, prove that (2 + √3) is an irrational number.

2

Seven times a two digit number is equal to four times the number obtained by reversing the order of its digits. If the difference of the digits is 3, determine the number.

3

Show that exactly one of the numbers n, n + 2 or n + 4 is divisible by 3.

4

If two positive integers p and q are written as p = a2b3 and q = a3b; a, b are prime numbers, then verify:
LCM (p, q) × HCF (p, q) = pq

5

Use Euclid’s algorithm to find the HCF of 4052 and 12576.

6

Using Euclid’s division algorithm find the HCF of the numbers 867 and 255.

7

Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

8

Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

9

A sweet seller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of that can be placed in each stack for this purpose?