1

Part of 2015 Tournament of Towns

Problems(2)

First digit of Powers of an Integer

Source: Tournament of Towns Spring 2015 Senior A-level

(a) The integers

x

x^2

x^3

begin with the same digit. Does it imply that this digit is

1

2

points)
(b) The same question for the integers

x, x^2, x^3, \cdots, x^{2015}

3

Relatively terms of a Geometric Series

Source: Tournament of Towns Fall 2015 Senior A-level

A geometrical progression consists of

37

positive integers. The first and the last terms are relatively prime numbers. Prove that the

19^{th}

term of the progression is the

18^{th}

power of some positive integer. (

3

number theoryalgebrageometric sequencegeometric series