Question
What did you notice sum of the digits of each multiple of 9
Answers
GPT-4o mini
The sum of the digits of each multiple of 9 has an interesting property: it is always a multiple of 9. This is a consequence of the divisibility rule for 9, which states that a number is divisible by 9 if the sum of its digits is also divisible by 9.
For example, consider the first few multiples of 9:
- \(9\): \(9\) (sum of digits = 9)
- \(18\): \(1 + 8 = 9\) (sum of digits = 9)
- \(27\): \(2 + 7 = 9\) (sum of digits = 9)
- \(36\): \(3 + 6 = 9\) (sum of digits = 9)
- \(45\): \(4 + 5 = 9\) (sum of digits = 9)
- \(54\): \(5 + 4 = 9\) (sum of digits = 9)
- \(63\): \(6 + 3 = 9\) (sum of digits = 9)
- \(72\): \(7 + 2 = 9\) (sum of digits = 9)
- \(81\): \(8 + 1 = 9\) (sum of digits = 9)
- \(90\): \(9 + 0 = 9\) (sum of digits = 9)
As you can see, the digit sums of the multiples of 9 also produce sums that themselves are multiples of 9, specifically equal to 9 for these examples.
This property holds for all multiples of 9, although the sum of the digits can be greater than 9 as the numbers increase, but will always yield a sum that is itself a multiple of 9. For instance, for the multiple \(99\), the sum of the digits is \(9 + 9 = 18\), which is also a multiple of 9. For \(108\), the sum is \(1 + 0 + 8 = 9\).
Overall, the sums continue to reflect the divisibility of the original multiples by 9 throughout the sequence.
For example, consider the first few multiples of 9:
- \(9\): \(9\) (sum of digits = 9)
- \(18\): \(1 + 8 = 9\) (sum of digits = 9)
- \(27\): \(2 + 7 = 9\) (sum of digits = 9)
- \(36\): \(3 + 6 = 9\) (sum of digits = 9)
- \(45\): \(4 + 5 = 9\) (sum of digits = 9)
- \(54\): \(5 + 4 = 9\) (sum of digits = 9)
- \(63\): \(6 + 3 = 9\) (sum of digits = 9)
- \(72\): \(7 + 2 = 9\) (sum of digits = 9)
- \(81\): \(8 + 1 = 9\) (sum of digits = 9)
- \(90\): \(9 + 0 = 9\) (sum of digits = 9)
As you can see, the digit sums of the multiples of 9 also produce sums that themselves are multiples of 9, specifically equal to 9 for these examples.
This property holds for all multiples of 9, although the sum of the digits can be greater than 9 as the numbers increase, but will always yield a sum that is itself a multiple of 9. For instance, for the multiple \(99\), the sum of the digits is \(9 + 9 = 18\), which is also a multiple of 9. For \(108\), the sum is \(1 + 0 + 8 = 9\).
Overall, the sums continue to reflect the divisibility of the original multiples by 9 throughout the sequence.