MathDB

Let

a

and

b

be integers. Define

a

to be a power of

b

if there exists a positive integer

n

such that

a = b^n

. Define

a

to be a multiple of

b

if there exists an integer

n

such that

a = bn

. Let

x

y

and

z

be positive integer such that

z

is a power of both

x

and

y

. Decide for each of the following statements whether it is true or false. Prove your answers. (a) The number

x + y

is even. (b) One of

x

and

y

is a multiple of the other one. (c) One of

x

and

y

is a power of the other one. (d) There exist an integer

v

such that both

x

and

y

are powers of

v

(e) For each power of

x

and for each power of

y

, an integer

w

can be found such that

w

is a power of each of these powers. (f) There exists a positive integer

k

such that

x^k > y

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