all the nubers from 1 to 99 are multiplied together. Use a pattern to determine the last digit of the product. Justify your answer.
all the mumbers from 1 to 99
clearly the last digit has to be a zero.
(actually the last few digits will be zeros, since factors like 2x5 or 10 or 20 etc all will produce an extra zero in the product)
To determine the last digit of the product when multiplying all the numbers from 1 to 99 together, let's analyze the pattern:
For the last digit of any number multiplied by another number, the resulting last digit is determined by the multiplication of the individual last digits. For example:
- When 2 is multiplied by any number, the last digit of the product will be the same as the last digit of the other number. (e.g., 2 x 3 = 6)
- When 3 is multiplied by any number, the resulting last digit is the number obtained by adding the last digit of the other number to itself. (e.g., 3 x 4 = 12, so the last digit is 2)
- When 4 is multiplied by any even number, the last digit of the product will be 0. (e.g., 4 x 2 = 8)
Based on these observations, we can determine the pattern for the last digit of multiplying numbers from 1 to 9:
1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 = 362,880
As you can see, the last digit of this product is 0. Therefore, when multiplying all the numbers from 1 to 99 together, the last digit of the product will also be 0.