How many times must you multiply a number's digits before reaching a single digit?
The multiplicative persistence of a number is the count of steps to reduce it to a single digit by repeatedly multiplying its digits together. For example, in base 10: 679 → 6×7×9 = 378 → 3×7×8 = 168 → 1×6×8 = 48 → 4×8 = 32 → 3×2 = 6, so 679 has persistence 5.
The available digits (0 through b−1) determine which products are possible at each step, completely reshaping the persistence landscape from base to base.
Any number containing a 0 digit immediately collapses to 0 (persistence 1 beyond the 0 step).
Colors indicate persistence level, from
0 (single digit) through
2,
3,
4,
5 and higher.