Show that there are no integer solutions m, n of the equation
æ ç
è
54
ö ÷
ø
m
=
æ ç
è
21
ö ÷
ø
n
which
gives the number of major thirds in an octave on a musical scale.
Given integers a, b, c and d where a and b are coprime and
c and d are coprime, find necessary and sufficient conditions
for there to exist positive integers m and n such that
æ ç
è
ab
ö ÷
ø
m
=
æ ç
è
cd
ö ÷
ø
n
.
[The set of three problems
Tuning and Ratio ,
Euclid's Algorithm and Musical Intervals and this problem
Rarity were devised by Benjamin Wardaugh who used to be a member
of the NRICH team. Benjamin is now doing research on the history
of music and mathematics at Oxford University. Read Benjamin's
article Music
and Euclid's Algorithm for more on this subject.]