All the numbers from 1 to 99 are multiplied together. What is the last digit of the product?
Zero
To find the last digit of the product of all numbers from 1 to 99, we need to understand the pattern of how the last digit changes as we multiply numbers together.
Let's observe the last digit of each number from 1 to 99:
1: 1
2: 2
3: 3
4: 4
5: 5
6: 6
7: 7
8: 8
9: 9
10: 0
11: 1
12: 2
...
19: 9
20: 0
...
From this pattern, we can see that the last digit repeats in cycles of 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. So if we multiply any number by a multiple of 10 (e.g., 10, 20, 30, etc.), the last digit of the product will always be 0.
Now, let's consider the multiplication of all numbers from 1 to 99. We can break it down into smaller steps to simplify the calculation:
1 * 2 * 3 * ... * 97 * 98 * 99
Notice that we have multiples of 10 in this sequence (10, 20, 30, ..., 90), and as we mentioned earlier, multiplying any number by a multiple of 10 results in a last digit of 0. So the product of all these multiples of 10 will have a last digit of 0.
Therefore, we can ignore these multiples of 10 and focus on the remaining numbers from 1 to 99. Let's remove the multiples of 10 from the multiplication:
1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 11 * 12 * ... * 97 * 98 * 99
By multiplying these remaining numbers, we can determine the last digit of the product.
Calculating the last digit of the product can be done by multiplying the last digits of each number sequentially. However, this process can be quite tedious and time-consuming. Instead, we can use some properties of modular arithmetic to simplify the calculation.
In modular arithmetic, we only care about the remainder of a number when divided by a certain base. In this case, we are interested in the remainder of each number when divided by 10, which gives us the last digit.
By multiplying the last digits of each number, we ignore all other digits and only focus on the last digit of each number. Because multiplication is commutative, the order in which we multiply the numbers does not matter.
Using this approach, we can calculate the product of the last digits of each number:
1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 1 * 2 * ... * 7 * 8 * 9
This can be simplified to:
(1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9)^9
Now, we can calculate the product of the last digits of the numbers 1 to 9, which is (1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9) = 362,880.
Taking the last digit of this number ((362,880 mod 10) = 0), we can conclude that the last digit of the product of all numbers from 1 to 99 is 0.
So, the answer is 0.