Given that is a rational number, what can you say about the value of n?
Only integer powers of 10 are rational. So the exponent
must be an integer, say k
n-1 = 2k
n = 2k+1
Therefore n must be odd in order that
be rational.
What if both and are rational? What can you say about the value of n?
For
Only integer powers of 10 are rational. So the exponent
must be an integer, say p
n-1 = 3p
n = 3p+1
Therefore n must be 1 more than a multiple of 3
Now we set the exponents equal:
n = 3p+1 = 2k+1
3p = 2k
2p+p = 2k
p + = 2k
= 2k-p
The right side is an integer, so the left side must be also
Let that integer be A
Then
= A
p = 2A
Substitute in
3p = 2k
3(2A) = 2k
6A = 2k
3A = k
Now supstitute in
n = 3p+1 = 2k+1
n = 3(2A)+1 = 2(3A)+1
n = 6A+1 = 6A+1
So n must be 1 more than a multiple of 6.
Edwin