How can I find cube root of a 9 digit number
How can I find cube root of a 9 digit number
We can easily calculate the first digit by considering the 1st 3 digit and last digits of the number by previous technique of a 9 digit number .
So, for example, looking at 580093704, we compare . The first 3 digit is 580 which is greater than 512(8^3).So first digit is 8
NUMBER CUBE LAST DIGIT OF CUBE 0 0 0 1 1 1 2 8 8 3 27 7 4 64 4 5 125 5 6 216 6 7 343 3 8 512 2 9 729 9
So the last digit of the cube root must be 4 as you have learned before.
So the next question is about middle digit.
To find out the middle digit we need a different module,11 module.
There is a short cut for modulo 11, but it is a little harder. Starting from the right end, you alternatively add and subtract digits. So if the number is 580093704. 580093704 = 4 – 0 + 7 – 3 + 9 – 0 + 0 – 8 + 5 = 14 = 4 – 1 = 3 (modulo 11). (In this case we luckily ended up with a positive number less than 11, but you might have to adjust the answer to a number in this range by adding or subtracting 11.)
To make use of these moduli, we need to calculate the table of cubes mod 11:
n n3 n3 mod 11 0 0 0 1 1 1 2 8 8 3 27 5 4 64 9 5 125 4 6 216 7 7 343 2 8 512 6 9 729 3 10 1000 10
we can see that each value from 0 to 10 occurs only once in the list of cubes modulo 11 (from 0 to 10), which means that cube roots can be calculated modulo 11.
Returning back to the worked example, 580093704 = 3 modulo 11, so (looking at the table, which, by the way, you will have to memorize) its cube root = 9 modulo 11. We’ve already determined that the first digit is 8 and the last digit is 4. If we think of the cube root as 8x4, for some x, then we have the equation 4 -x + 8 = 9 modulo 11, i.e. 12 - x = 9 modulo 11, so x = 3 modulo 11. Which gives a final (and correct) answer of 834.
We can easily calculate the first digit by considering the 1st 3 digit and last digits of the number by previous technique of a 9 digit number .
So, for example, looking at 580093704, we compare . The first 3 digit is 580 which is greater than 512(8^3).So first digit is 8
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 2 | 8 | 8 |
| 3 | 27 | 7 |
| 4 | 64 | 4 |
| 5 | 125 | 5 |
| 6 | 216 | 6 |
| 7 | 343 | 3 |
| 8 | 512 | 2 |
| 9 | 729 | 9 |
So the last digit of the cube root must be 4 as you have learned before.
So the next question is about middle digit.
To find out the middle digit we need a different module,11 module.
There is a short cut for modulo 11, but it is a little harder. Starting from the right end, you alternatively add and subtract digits. So if the number is 580093704. 580093704 = 4 – 0 + 7 – 3 + 9 – 0 + 0 – 8 + 5 = 14 = 4 – 1 = 3 (modulo 11). (In this case we luckily ended up with a positive number less than 11, but you might have to adjust the answer to a number in this range by adding or subtracting 11.)
To make use of these moduli, we need to calculate the table of cubes mod 11:
| n | n3 | n3 mod 11 |
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 2 | 8 | 8 |
| 3 | 27 | 5 |
| 4 | 64 | 9 |
| 5 | 125 | 4 |
| 6 | 216 | 7 |
| 7 | 343 | 2 |
| 8 | 512 | 6 |
| 9 | 729 | 3 |
| 10 | 1000 | 10 |
we can see that each value from 0 to 10 occurs only once in the list of cubes modulo 11 (from 0 to 10), which means that cube roots can be calculated modulo 11.
Returning back to the worked example, 580093704 = 3 modulo 11, so (looking at the table, which, by the way, you will have to memorize) its cube root = 9 modulo 11. We’ve already determined that the first digit is 8 and the last digit is 4. If we think of the cube root as 8x4, for some x, then we have the equation 4 -x + 8 = 9 modulo 11, i.e. 12 - x = 9 modulo 11, so x = 3 modulo 11. Which gives a final (and correct) answer of 834.
