# What is a Decomposed Word™?

This is in the spirit of the What is a Word/Phrase™ series started by JLee with a special brand of Phrase™ and Word™ puzzles.

If a word has a certain property, I call it a Decomposed Word™.

You can use the examples below to find the property:

$$\begin{array}{|c|c|} \hline \bbox[yellow]{\textbf{Decomposed Words™}}&\bbox[yellow]{\textbf{Not Decomposed Words™}}\\ \hline \text{JOINING}&\text{ADHERING}\\ \hline \text{AGGRAVATE}&\text{EXACERBATE}\\ \hline \text{ANNEXING}&\text{APPENDING}\\ \hline \text{WATCHLIST}&\text{REGISTER}\\ \hline \text{VANQUISHING}&\text{DEFEATING}\\ \hline \text{PLAINTEXT}&\text{CLEARTEXT}\\ \hline \text{DESEXUALIZE}&\text{EMASCULATE}\\ \hline \text{MOCCASIN}&\text{SLIPPER}\\ \hline \text{LITIGATION}&\text{LAWSUIT}\\ \hline \text{MIX}&\text{STIR}\\ \hline \text{CODIFY}&\text{TABULATE}\\ \hline \text{METHAMPHETAMINE}&\text{OPIOID}\\ \hline \text{CONQUERING}&\text{SUBJUGATING}\\ \hline \end{array}$$

For those without MathJax, or if you want to pop this into a spreadsheet, here is a CSV version:

Decomposed™,Not Decomposed™
AGGRAVATE,EXACERBATE
ANNEXING,APPENDING
WATCHLIST,REGISTER
VANQUISHING,DEFEATING
PLAINTEXT,CLEARTEXT
DESEXUALIZE,EMASCULATE
MOCCASIN,SLIPPER
LITIGATION,LAWSUIT
MIX,STIR
CODIFY,TABULATE
METHAMPHETAMINE,OPIOID
CONQUERING,SUBJUGATING

• Possible duplicate of What is a Composite Word™? – Rand al'Thor Sep 26 '16 at 17:11
• @randal'thor I would argue it's different, since one has to do with the sum of the contained RNs, while the other contains only a single RN. We can let the community decide, though. – GentlePurpleRain Sep 26 '16 at 17:13
• @randal'thor I was not aware of the other question when I posted mine (or I probably wouldn't have posted it). – GentlePurpleRain Sep 26 '16 at 17:14
• Yes, there's a very slight difference, but in most cases the two would come to the same thing. And the idea needed - the interesting part - is exactly the same ("think of Roman numerals"). They're both worthy of upvotes though, of course :-) – Rand al'Thor Sep 26 '16 at 17:17

I think Decomposed Words are

those for which, if you extract the letters that are used for Roman numerals, make a prime number

and Not Decomposed Words are accordingly

those which yield either a composite number or 1.

The given examples don't let us tell

how to classify words that contain no such letters.

Per GentlePurpleRain's suggestion in comments: the name is because

when you decompose a number as far as possible into factors, the components you end up with are the prime numbers (what about 1? I hear you cry; well, when you decompose 1 into factors you get the empty set of factors, all of whose elements are in fact prime).

And here are the actual values:

 2 II JOINING | 501 DI ADHERING 5 V AGGRAVATE | 90 XC EXACERBATE 11 XI ANNEXING | 501 DI APPENDING 151 CLI WATCHLIST | 1 I REGISTER 7 VII VANQUISHING | 501 DI DEFEATING 59 LIX PLAINTEXT | 160 CLX CLEARTEXT 541 DXLI DESEXUALIZE | 1150 MCL EMASCULATE 1201 MCCI MOCCASIN | 51 LI SLIPPER 53 LIII LITIGATION | 51 LI LAWSUIT 1009 MIX MIX | 1 I STIR 401 CDI CODIFY | 50 L TABULATE 3001 MMMI METHAMPHETAMINE | 498 IID OPIOID 101 CI CONQUERING | 1 I SUBJUGATING

You may notice that the second-last one of those on the right

is nonstandard because you can't really put two smaller "decrementing" Is before a single D. DII is also composite (of course; it's even and doesn't equal 2) but if we just add up Roman numerals without regard to position then MIX no longer works because MXI = 1011 which, being a multiple of 3, is not prime.

But the questioner indicates in comments that

his intention was that anything not containing a valid Roman numeral should be a Not Decomposed Word, so OPIOID is fine for that reason.

• those words are Decomposing – Sconibulus Sep 19 '16 at 20:21
• Perhaps you could flesh out your answer by providing a) some examples using the words provided above and b) an explanation of why they are called "decomposed"? – GentlePurpleRain Sep 19 '16 at 20:26
• @GentlePurpleRain, done. I'm a little puzzled by OPIOID, but perhaps I'm just on drugs or something. – Gareth McCaughan Sep 19 '16 at 21:15
• Wow! It wasn't my intention for the non-decomposed words to even generate anything valid at all. I'm surprised that (just by chance) all but one of them did... – GentlePurpleRain Sep 19 '16 at 21:17
• Ah, OK. Then I am no longer puzzled by OPIOID. – Gareth McCaughan Sep 19 '16 at 21:24