If a Solar Day can't fit into a Solar Year, that is a huge problem. Where are those extra hours coming from?
If a Solar Day is how long it takes the earth to rotate on its axis.
And a Solar Year is how long it takes the earth to orbit the sun.
Why must the ratio of those be an integer? That implies that for every orbit the earth makes of the sun it rotates exactly 'x' times, and x is an integer.
Why should it be? The rotation speed (which, by the way, does change over time, very slowly) and orbit speed are determined by all kinds of things, there is no reason the ratio should be an integer.
Division of the number of days in a year and the number of hours in a day ( Days in a Year / Hrs in a Day ) must be a whole number because each Day in the Year is a representative of 24 Hours. If this is true then the ratios must relate.
Imagine that we had a planet with a Solar Day that was a 10 hour day.
Imagine that that the Solar Year year of this planet was 100 days.
Lets define Solar Year as the time it takes for the Sun to return back to the same place in the sky in Solar Time. This means that each of the days in the year must be full rotations.
Does a Solar Day fit into a Solar Year?
100 / 10 = 10. Yes. A solar day fits into a Solar Year.
If we mess around those numbers, a solar day no longer fits into a Solar Year.
142 / 10 = 14.2. A solar day does not fit into a Solar Year.
100 / 7 = 16.6. A solar day does not fit into a Solar Year.
A result of a whole number shows that the second value fits into the first value. The only types of Years a 10 hour day would return whole numbers in such a division.
A 10 hour day can fit into a 10 day year (10 / 10 = 1), a twenty day year ( 20 / 10 = 2 ), but not a 25 day year ( 25 / 10 = 2.5 ).
Each 10 doesn't fit nicely into the 25, and the result is 2.5. The Solar Days have not been completed by the time the planet makes its way to the end.
It doesn't make a difference if we call them planets and years or dimes and pennies. The relationship is defined and must be maintained.
Relationship: 100 pennies is 10 dimes. There are 10 pennies in a dime.
We have 142 pennies and 10 dimes. Does the relationship work?
142 pennies / 10 dimes = 14.2. This is not a whole number. 10 dimes does not fit into a value that is 142 pennies.
100 pennies / 7 dimes = 16.2. This is not a whole number. 7 dimes does not fit into a value that is 100 pennies.
10 dimes can fit into 10 pennies (10 / 10 = 1), 20 pennies (20 / 10 = 2), but it cannot fit into 25 pennies (25 / 10 = 2.5).
10 dimes doesn't fit nicely into the 25, and the result is 2.5.