So, i'd like to offer three new words ~ new to me, at any rate; probably someone has previously used and defined them ~ which the language seems to lack. They are formed along the lines of odd and even, as you'll see: They are treven (pronounced 'tree-vən), trod, and trud. I define them like this: A number which is perfectly divided by three (i.e., with no remainder) is treven; a number which, when divided by three leaves a remainder of one is trod; obviously, a number which divided by three gives a remainder of two is trud. (As a mnemonic, a trod number is one over a treven number, a trud number is one under.) As i say, these, or others with similar meanings, may well have been created and used for years, by mathematicians if no one else, but i have not come across them, and found the need for them in thinking about some ideas today.
Everyone knows that a number divisible by three has digits that add to three or to another number divisible by three. For example, 345 is evenly divisible by three because 3+4+5=12 and 12 is divisible by three because 1+2=3. Thus it is clear that a number is treven if its digits add to 3 or 6 or 9 or any other treven number.
So i wondered, is there a way to tell, if a number is not treven, is it trod or trud? Well, let's look at a few examples. 749÷3=249 remainder 2; it is trud, and 7+4+9=20, which is a trud number, and 2+0=2, which is also trud. 157942=52647 remainder 1; it is trod, and 1+5+7+9+4+2=28, a trod number, and 2+8=10, also a trod number. 6847004 is trud; 6+8+4+7+4=29, which is also trud; 2+9=11, trud again, and 1+1=2, also trud.
There are seven example numbers (749, 20, 157942, 28, 6847004, 29, 11) seeming to follow a pattern which suggests itself: Just as a treven number's digits add to a treven number, a trod number's digits will add to a trod number, and those of a trud number to a trud number. A lovely, clear pattern, something i love to find in numbers. But, along the lines of single swallows and summer, four numbers do not a law make. Therefore, i need someone clever to tell me, is this a known law? And, more importantly, why does it work?