Least Significant Digits

Copyright 2003 by Paul Niquette.  All rights reserved.

While laboring over the clues for the first Cross-Number Puzzle, a certain puzzle writer fashioned an inventory of interesting integers from which to make selections.

Primes and powers, factorials and Fibonacci numbers, these all became candidates for the puzzle.  Also quite interesting are the Sums of the First n Integers, but they were not tabulated, for as we learned in Roulette's Frets, each sum, no matter how large, can be readily calculated as needed.  But wait, there was a problem with the Least Significant Digits in those sums. 
Some ciphers, it seems, are extremely rare, which can be mighty inconvenient when one is creating a puzzle. 
Rare?  No, they were missing altogether.  An exclamation point may be appropriate, inasmuch as a discovery is at hand.  Not a big discovery, maybe, but -- hey this is the kind that applies a particular power of perception.  And it takes practice. 
Oh sure, discoveries are conventionally made by looking and listening, thereby seeing and hearing what's there to be seen and heard.  The sophisticated solver, though, will recall one of the most famous Sherlock Holmes stories...

 

"A dog was kept in the stables, and yet, though someone had been in and had fetched out a horse, he had not barked enough to arouse the two lads in the loft. Obviously the midnight visitor was someone whom the dog knew well."
 

-- from the short story "Silver Blaze"
by Sir Arthur Conan Doyle
To catch a thief, the sophisticated detective must listen for what is not there to be heard.  To make a discovery, the sophisticated solver must look for what is not there to be seen.
Thus, by observing the absence of certain decimal ciphers as the Least Significant Digits in sums of consecutive integers, the sophisticated solver might well be made to wonder...
What is the explanation?

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