number-of-positive-integers-a-and-b-and-c-satisfying-a-b-c-b-c-a-c-a-b-5abc-

Question Number 16053 by vpawarksp@gmail.com last updated on 17/Jun/17

n u m b e r o f p o s i t i v e i n t e g e r s a a n d b a n d c s a t i s f y i n g

a^{b^{c}} b^{c^{a}} c^{a^{b}} = 5 a b c

Commented by prakash jain last updated on 17/Jun/17

Given a, b, c are integers

at least one of a, b, c is a multiple of

5 .

Let a = 5

If a = 5, RHS = 25 b c

c a s e 1 :

\Rightarrow b^{c} = 2 o r b = 2, c = 1

5^{2} 2^{1} 1^{25} = 50

c a s e 2 :

if a = 5 and RHS is a muliple of 25

say b = 5

a^{b^{c}} increases at much faster rate

with c than 125 c .

s o n o s o l u t i o n s w i t h a = 5, b = 5

- - - - -

s i m i l a r l y f o r b = 5 a n d c = 5

s o l u t i o n s

a = 5, b = 2, c = 1

b = 5, c = 2, a = 1

c = 5, a = 2, b = 1