Persistence of a number assumed. The single-digit final state reached in the process of calculating an integer's additive persistence is its digital root. Put another way, a number's additive persistence counts how many times we must sum its digits to arrive at its digital root. The additive persistence of 2718 is 2: first we find that 2 + 7 + 1 + 8 = 18, and then that 1 + 8 = 9. The multiplicative persistence of 39 is 3, because it takes three steps to reduce 39 to a single digit: 39 → 27 → 14 → 4. Also, 39 is the smallest
1089 (number) A − C ≤ 9, both the first and third digits are guaranteed to be single digits.) Its reversal is 100 × ['] + 10 × ' + 1 × [']. The sum is thus 101 × ['] + 20 × ' + 101 × ['] = 101 × [' + '] + 20 × = 101 × [−1 + 10] + 180 = 1089. Multiplying the number 1089 by the integers from 1 to 9 produces a pattern: multipliers adding up to 10 give products that are the digit reversals of each other: Also note the patterns within each