All even numbers from 12 to 98 inclusive, except those ending in 0, are…

Concept: The units digit of a product depends only on the units digits of its factors, and repeated multiplication by numbers sharing the same last-digit pattern follows a cyclicity — so a large product's units digit can be found by combining the units digits of manageable chunks and then applying the cyclicity of powers, rather than multiplying every term out.

Applying this to the given product:

1. List every even number from 12 to 98 inclusive that does not end in 0: 44 even numbers exist in this range and 8 of them (20, 30, 40, 50, 60, 70, 80, 90) end in 0, leaving 36 qualifying numbers.

2. Group these 36 numbers into 9 sets of four consecutive even numbers, one set per decade: (12, 14, 16, 18), (22, 24, 26, 28), (32, 34, 36, 38), and so on through (92, 94, 96, 98).

3. Every set has the same units digits — 2, 4, 6, 8 — because the tens digit doesn't affect the units digit. Their product is 2 × 4 × 6 × 8 = 384, so each set contributes a units digit of 4 to the running product.

4. With 9 such sets multiplied together, the overall units digit equals the units digit of 49, since multiplying nine numbers that each end in 4 compounds the units digit the same way as raising 4 to the 9th power.

5. Powers of 4 cycle with period 2: 41 = 4, 42 = 16 (units digit 6), 43 = 64 (units digit 4), and so on — odd powers end in 4, even powers end in 6.

6. Since 9 is odd, 49 ends in 4.

Cross-check: multiplying just two sets' worth of the pattern gives 4 × 4 = 16, ending in 6 — matching 42 as predicted by the cyclicity above. Extending the same cycle to 9 sets (an odd count) confirms the units digit is 4.

So the rightmost (units) digit of the product is 4.