Hello, Bartimaeus!

I assume you are somewhat familiar with base-12 notation.

To work in base-12, we must invent new symbols for digits 10 and 11.

Use $\displaystyle t$ for ten and $\displaystyle e$ for eleven. For example, $\displaystyle 2 \times 5 = t_{12}$

(a) Write out 3-, 5- and 8-times table in base-12, up to $\displaystyle e$-times. Code:

x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | t | e |
-- + -- + -- + -- + -- + -- + -- + -- + -- + -- + -- + -- +
3 | 3 | 6 | 9 | 10 | 13 | 16 | 19 | 20 | 23 | 26 | 29 |
-- + -- + -- + -- + -- + -- + -- + -- + -- + -- + -- + -- +
5 | 5 | t | 13 | 18 | 21 | 26 | 2e | 34 | 39 | 42 | 47 |
-- + -- + -- + -- + -- + -- + -- + -- + -- + -- + -- + -- +
8 | 8 | 14 | 20 | 28 | 34 | 40 | 48 | 54 | 60 | 68 | 74 |
-- + -- + -- + -- + -- + -- + -- + -- + -- + -- + -- + -- +

(b) In the times tables in (a), some have answers with the last digit 0.

Which tables in base-10 [up to 9-times] have answers with the last digit 0?

In base-10, the rows that have answers ending in 0 are:

. . 2-times, 4-times, 6-times, 8-times, and 5-times.

(c) Explain why some tables have answers with last digit 0 and others do not

in base-10 [up to 9-times] and in base-12 [up to *e* times]

For base-ten: $\displaystyle 10 = 2\cdot5$

Any row which shares a common factor with 10 has answers ending in 0.

Any row which is relative prime to 10 has __no__ answers ending in 0.

For base=twelve: $\displaystyle 12 = 2^2\cdot3$

Any row which shares a common factor with 12 has answers ending is 0.

. . They are: 2-times, 3-times, 4-times, 6-times, 8-times, 9-times, and $\displaystyle t$-times

(d) Which tables in base-12 [up to $\displaystyle e$-times] will have no answers with last digit 0?

Any row which is relatively prime to 12 has __no__ answers ending in 0.

. . They are: 5-times, 7-times, and $\displaystyle e$-times.