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

Express each number as a product of its prime factors:

(i) 140

(ii) 156

(iii) 3825

(iv) 5005

(v) 7429

Given that HCF (306, 657) = 9, find LCM (306, 657).

Prove that 3 + 2√5 is irrational

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(i) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600 

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form, p/q what can you say about the prime factors of q?

(i) 43.123456789

(ii) 0.120120012000120000. . .

Check whether 6n can end with the digit 0 for any natural number n.

What is the HCF of the smallest prime number and the smallest composite number?

Using Euclid’s Algorithm, find the HCF of 2048 and 960.

Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two given numbers.

Find three rational numbers lying between 0 and 0.1. Find twenty rational numbers between 0 and 0.1. Give a method to determine any number of rational numbers between 0 and 0.1.