Q1.
If this difference is 11 or a multiple of 11, what does that say about the remainder obtained when the number is divisible by 11?
Answer
The remainder is 0 — the number is exactly divisible by 11.
In the book's example, 320185:
excess = 2 + 1 + 5 = 8, short = 3 + 0 + 8 = 11
8 – 11 = – 3 → 3 short of a multiple of 11
excess = 2 + 1 + 5 = 8, short = 3 + 0 + 8 = 11
8 – 11 = – 3 → 3 short of a multiple of 11
Now suppose the difference had come out as 11, or 22, or – 11. That leftover is itself a whole number of 11s, so it joins the multiple of 11 already built up and nothing at all remains.
Why it happens: the number equals (a multiple of 11) + (the difference). If the difference is also a multiple of 11, the total is a multiple of 11 plus a multiple of 11 — still a multiple of 11, by the rule that a | M and a | N gives a | M + N.
Tip: if the difference is negative, add 11 (or a multiple of 11) to it to read off the remainder. A difference of – 3 means remainder 11 – 3 = 8.