Write the denominator of rational number in the form 2^m × 5ⁿ, where m, n are non-negative integers. Hence, write its decimal expansion, without actual division.

Show that 12n cannot end with the digit 0 or 5 for any natural number n.

On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm, and 45 cm, respectively. What is the minimum distance each should walk, so that each can cover the same distance in complete steps?

Prove that is irrational, where p and q are primes.

Show that the cube of a positive integer of the form 6q + r, q is an integer and r = 0,1,2,3,4,5 is also of the form 6m + r.