Question

The number of 15th roots of unity which are also the 25th roots of unity?
Can we generalise a way for doing this.

Answer 1

do you mean common roots of 15th and 25th??

Answer 2

Yes.

Answer 3

Roots of 15 =
\[ \cos \frac{2\pi k }{15} + i \sin \frac{2\pi k }{15}\]

Roots of 25 =
\[ \cos \frac{2\pi k }{25} + i \sin \frac{2\pi k }{25}\]

Answer 4

k ranges from 1 to 15 in 15th root and 1 to 25 in 25th

Answer 5

is there a possible common root other than k=15 and k=25 ??

Answer 6

for integer values of k ... let me check for programming solution

Answer 7

apparently there seems to be
3/15 = 5/25

6/15 = 10/25

12/15 = 20/25

15/15 = 25/ 25

Answer 8

Hmmmm.
Now it occurs to me that if we plot it on the unit circle the we get solutions of arguments : \[2\pi k/n1 \]
And
\[2\pi k'/n2\]
Type.
Each complex number has a modulus one.
So basically what we need to do for finding coincident roots is that take the LCM of the given arguments and all the multiples of that between 0 to 2pi should be the common ones.
\[LCM [2\pi k/n1 ,2\pi k'/n2] * m < 2\pi \]
All the values that satisfy this should be correct.
I think.
Does it make sense?

Answer 9

Ah i'm starting to have head ache now ... lol
it seems to be to this form
1/5, 2/5, 3/5, 4/5, 5/5

Answer 10

Lol.
Exactly.
And LCM of 1/25 and 1/15 is 1/5.
So it consists of all the multiples of 1/5 uptil 2pi.

Answer 11

25 has only two factors 5,5 <-- and only when denominator is less than 25 it's possible to ave common factors between 25 and 15 ... since 15 is also divisible by 5, this is possible to hav five factors.

Answer 12

Jeez ... that's HCF

Answer 13

Ohhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh.
Sorry. My bad. I tend to mess up terms a lot. :P

Answer 14

no worries ... still you managed to get answer without assistance of computer!!

Answer 15

Haha. :P Some rare times. :)