MathDB

Let

a

b

c

and

n

be positive integers such that

a^n

is divisible by

b

, such that

b^n

is divisible by

c

, and such that

c^n

is divisible by

a

. Prove that \left(a \plus{} b \plus{} c\right)^{n^2 \plus{} n \plus{} 1} is divisible by

abc

. An even broader generalization, though not part of the QEDMO problem and not quite number theory either: If

u

and

n

are positive integers, and

a_1

a_2

, ...,

a_u

are integers such that

a_i^n

is divisible by a_{i \plus{} 1} for every

i

such that

1\leq i\leq u

(we set a_{u \plus{} 1} \equal{} a_1 here), then show that \left(a_1 \plus{} a_2 \plus{} ... \plus{} a_u\right)^{n^{u \minus{} 1} \plus{} n^{u \minus{} 2} \plus{} ... \plus{} n \plus{} 1} is divisible by

a_1a_2...a_u

Problem Statement