The problem below seems fairly easy... but I just can't manage to prove it!
PROBLEM: Given posistive integers a and b such that a|b^2, b^2|a^3, a^3|b^4, b^4|a^5, ... , prove that a = b.
The problem below seems fairly easy... but I just can't manage to prove it!
PROBLEM: Given posistive integers a and b such that a|b^2, b^2|a^3, a^3|b^4, b^4|a^5, ... , prove that a = b.
this already tells us that every prime that divides a divides b and every prime that divides b divides a, so a,b are divisible by exactly the same primes. Well, now just prove that if a prime divides a to some power, then at that same power that primes divides b, and the other way around.
Tonio
quote=swallenberg;446066]But how? I came this far when I tried the first time... but I just can't seem to cross the finish line.[/quote]
Suppose and these powers are maximal in each case, i.e. , and suppose , then:
. But . Continue from here comparing odd powers of a with even powers of b.
Tonio