All the numbers from 1 to 99 are multiplied together. Use a pattern to determine the last digit of the product. Justify your answer.
It's a very simple pattern, really. :-)
Think what happens to the last digit of any number when multiplied by ten.
im sorry but i do not understand :(
Take any digit, say 2.
Multiply it by 10.
What's the last digit?
Multiply that by anything else. What's the last digit?
To determine the last digit of the product of all the numbers from 1 to 99, we can observe a pattern. Let's break it down step-by-step:
1. Start with the number 1.
2. Multiply it by 2, which gives us 2.
3. Multiply the result by 3, which gives us 6.
4. Multiply the result by 4, which gives us 24.
5. Multiply the result by 5, which gives us 120.
6. Multiply the result by 6, which gives us 720.
7. Continue this pattern until we reach 99.
By following this pattern, we can notice the following:
- Every time we multiply by 5, the last digit becomes 0 because 5 ends in 5, so any multiple of 5 will have a last digit of 0 (e.g., 10, 15, 20, etc.).
- Every time we multiply by 2, the last digit doubles until it reaches 10 (e.g., 2, 4, 6, 8, 10, 12, ...).
- Since there are more multiples of 2 than multiples of 5 in the range from 1 to 99, we can conclude that the overall product will have a last digit greater than 0.
- The last digit will only become 0 when we directly multiply by 5.
Therefore, the last digit of the product of all the numbers from 1 to 99 is 0.