Calculators
Multiplicative Persistence Calculator
Count repeated digit multiplications with exact step-by-step products and the final single digit, privately in your browser.
Repeatedly multiply decimal digits until one digit remains. Enter a non-negative integer of up to 1000 digits. Calculations stay in your browser.
How multiplicative persistence works
Multiplicative persistence counts how many times you multiply the decimal digits of a non-negative integer before reaching a single digit. For 39, the steps are 3 × 9 = 27, 2 × 7 = 14, and 1 × 4 = 4: persistence 3 and final digit 4. A single digit, including zero, needs zero rounds. A multi-digit number containing zero reaches zero in one round. Leading zeros are removed before counting, so 0007 is treated as 7. Unlike the digital root, which repeatedly adds digits, this calculation multiplies them. Enter 1 to 1000 digits; surrounding whitespace is accepted but signs, decimal points, separators and scientific notation are rejected. Exact BigInt arithmetic preserves every product digit without floating-point rounding. Each step is displayed and can be copied. All calculations run locally in your browser. This tool uses base 10.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is multiplicative persistence?
It is the number of digit-product rounds required to reach one digit. For 39, the sequence is 39 → 27 → 14 → 4, so the persistence is 3.
What happens with zero or a single digit?
Any single digit has persistence zero. A multi-digit number containing a zero reaches zero after one multiplication round.
Do leading zeros count?
No. Leading zeros are removed first, so 00039 has the same persistence as 39 and 0007 needs zero rounds.
Is this the same as the digital root?
No. A digital root repeatedly adds digits. Multiplicative persistence counts repeated multiplication rounds and also shows the final product digit.
Are large integers rounded or uploaded?
No. Inputs of up to 1000 digits use exact BigInt arithmetic entirely in your browser.