Answer:
Every 5n mark on the 385 scale will line up with every 6n mark on the 462 scale for n an integer from 0 to 77.
Starting at 0, the first set of marks to match are 5 on the 385 scale and 6 on the 462 scale.
Step-by-step explanation:
The numbers factor as ...
385 = 5×7×11
462 = 2×3×7×11
Their greatest common factor is 7×11 = 77, so there will be 77 marks (other than 0) that line up. On the 385 scale, every 385/77 = 5th mark will match up with every 462/77 = 6th mark on the 462 scale.
A few of the matching marks are ...
(385 scale, 462 scale) ⇒
(0, 0), (5, 6), (10, 12), (15, 18), ..., (375, 450), (380, 456), (385, 462)
_____
Another way to think of this is to consider the fractions ...
1/385, 2/385, 3/385, 4/385, 5/385, 6/385, ...
and compare them to the fractions ...
1/462, 2/462, 3/462, 4/462, 5/462, 6/462, 7/462, ...
You find that 5/385 = 6/462 = 1/77. That is, 5 units on the 385 scale represent 1/77 of the length of the scale, as do 6 units on the 462 scale. Multiples of 1/77 of each scale will be where the marks line up.
