what is the greatest number of times you would regroup when multiplying two 2-digit factors?
To find the greatest number of times you would regroup when multiplying two 2-digit factors, let's understand the process of regrouping in multiplication.
When multiplying two 2-digit factors, you typically perform multiplication in a vertical format, starting from right to left. Each digit in the second factor is multiplied by each digit in the first factor, and the results are then added together. The regrouping occurs when the product of two digits exceeds 9, resulting in a carry-over to the next column.
To determine the greatest number of times you would regroup, consider the scenario that would result in the most carry-overs. In multiplication, the largest product you can get from multiplying two single-digit numbers is 81 (9 x 9). To achieve this maximum product, you need the two digits being multiplied to be 9 each. Therefore, if you multiply two 2-digit numbers, and each of the digits in both numbers is 9, you will have the highest number of regroupings.
For example, let's consider multiplying 99 by 99:
99
x 99
__________
891 <- result of multiplying the last digits (9 x 9)
+ 8910 <- result of multiplying the tens digits (9 x 9), regrouped by carrying over
__________
9801 <- final result
In this case, there are two regroupings, one for the ones place and one for the tens place.
Therefore, the greatest number of times you would regroup when multiplying two 2-digit factors is two.