All the numbers from 1 to 99 are multiplied together. Use a pattern to determine the last digit of the product. Justify your answer.
To determine the last digit of the product of all the numbers from 1 to 99, we can analyze the pattern of the last digits.
1. The last digit of 1 is 1.
2. The last digit of 2 is 2.
3. The last digit of 3 is 3.
4. The last digit of 4 is 4.
5. The last digit of 5 is 5.
6. The last digit of 6 is 6.
7. The last digit of 7 is 7.
8. The last digit of 8 is 8.
9. The last digit of 9 is 9.
10. The last digit of 10 is 0.
11. The last digit of 11 is 1.
12. The last digit of 12 is 2.
...
From this pattern, we can see that the last digit repeats every 10 numbers because the pattern of the last digit of each number ends with 0, 1, 2, ... 9 and then repeats. Therefore, the last digit of every tenth number will be the same. This means that the last digit of the product of all the numbers from 1 to 99 will be the same as the last digit of the product of the numbers from 1 to 9.
The product of the numbers from 1 to 9 is 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 = 362,880. The last digit of this product is 0.
So, the last digit of the product of all the numbers from 1 to 99 is 0.
To determine the last digit of the product of all the numbers from 1 to 99, we can notice a pattern and justify our answer.
Let's consider the last digit of each number from 1 to 99:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
By observing the pattern, we can see that the last digit repeats in cycles of 10. The cycle starts with 1, goes up to 9, and then repeats again starting with 0. This pattern continues for all the numbers from 1 to 99.
Now, since the multiplication operation only affects the last digit, we can focus solely on the last digits of the numbers being multiplied. We can ignore the remaining digits, as they don't contribute to the last digit of the final product.
Let's examine the multiplication of the last digits for the numbers from 1 to 10:
(1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 0) = 0
As you can see, when we multiply the last digits 1 through 10, the product ends with a 0.
Now, let's consider the multiplication of the last digits for the numbers from 11 to 20:
(1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 0) = 0
The last digit of this product is also 0.
We can continue this process and realize that for each cycle of 10 numbers, the product of their last digits will end with 0. Since there are ten cycles of ten numbers in the range from 1 to 99, the products of their last digits will have a total of ten 0's at the end.
Hence, the last digit of the product of all the numbers from 1 to 99 is 0.
I do not know how to figure out the pattern, but when you do you can justify your answer by saying multiplying all the numbers together is exactly the problem "99!" which you can punch in to a calculator. I hope this is right and makes sense :/
1 + 99
2 + 98
3 + 97
...............
...............
49 51
+ 50
49(100) + 50