What is the 43rd digit after the decimal point in the decimal representation of 1/13?
1/13
= .076923076923076923...
notice the repeat is 076923 is 6 digits long
1st after decimal is 0
7th after the decimal is 0
13th after the decimal is 0
...
37th after the decimal is 0
43rd after the decimal is 0
notice that the numbers
1,7,13, ... 37, 43 all leave a remainder of 1 after dividing by 6, the length of my repeat
To find the 43rd digit after the decimal point in the decimal representation of 1/13, we need to perform long division. Here's how you can do it:
1. Start by dividing 1 by 13:
- The quotient is 0, and the remainder is 1. Write down the quotient (0) as the first digit after the decimal point.
2. Multiply the remainder (1) by 10 and divide by 13:
- The quotient is 0, and the remainder is 10. Write down the quotient (0) as the second digit after the decimal point.
3. Repeat step 2 for the desired number of times, keeping track of the remainders:
- Multiply the remainder (10) by 10 and divide by 13:
- The quotient is 0, and the remainder is 10. Write down the quotient (0).
- Multiply the remainder (10) by 10 and divide by 13:
- The quotient is 0, and the remainder is 10. Write down the quotient (0).
- Continue this pattern until you have reached the desired number of digits after the decimal point.
4. Finally, after performing the division 43 times, the 43rd digit after the decimal point is 7.
Therefore, the 43rd digit after the decimal point in the decimal representation of 1/13 is 7.