Dear Teacher:
The gcd (203, 161) = 7 .
As well, Euclid's algorith allows me to write 203:161 as a continued fraction:
203:161 =
= 1 + 42:161 =
= 1 + 1:(161:42) =
= 1 + 1:(3+35:42) =
= 1 + 1:(3+1:(42:35)) =
= 1 + 1:(3+1:(1+7:35))) =
= 1 + 1:(3+1:(1+1:5))) =
= [ 1; 3, 1, 5 ]
As well, Euclid's algorith allows me to divide a 203 x 161 rectangle into squares:
One 161 x 161 square
Three 42 x 42 squares
One 35 x 35 square
Five 7 x 7 squares
In total there are 1+3+1+5 = 10 squares...
As well, Euclid's algorithm allows me to calculate the gcd of two polynomials (even if Ruffini's rule isn't viable)...
So, gcd ( x5 + 2x4 + 6x3 + 3x2 + 8x – 2 , x4 + x3 + 5x2 + 2x + 6 ) = x2 + 2.
Pepe continued talking about continued fractions...
Dear Teacher:
Look at these periodic continued fractions:
The golden ratio [ 1; 1, 1, 1, 1 ... ] = (1+√5)/2
The silver ratio [ 2; 2, 2, 2, 2 ... ] = 1+√2
The bronze ratio [ 3; 3, 3, 3, 3 ... ] = (3+√13)/2
These metallic ratios are solutions to these equations:
Golden ratio 1+1/x = x.
Silver ratio 2+1/x = x.
Bronze ratio 3+1/x = x.
Because if we rationalize:
1+2/(1+√5) = 1+(√5–1)/2 = (1+√5)/2.
2+1/(1+√2) = 2+(√2–1)/1 = 1+√2.
3+2/(3+√13) = 3+(√13–3)/2 = (3+√13)/2.
The golden ratio is found in many canvases of great painters...