The sum of all digits that appear in the integers from 10 to 100 is 856.
Let's break this down:
We need to consider all numbers from 10 up to and including 100. We can look at the tens place and the ones place separately.
Tens Place:
- From 10-19, the tens digit is always 1. This appears 10 times. The sum is 1 * 10 = 10.
- From 20-29, the tens digit is always 2. This appears 10 times. The sum is 2 * 10 = 20.
- From 30-39, the tens digit is always 3. This appears 10 times. The sum is 3 * 10 = 30.
- From 40-49, the tens digit is always 4. This appears 10 times. The sum is 4 * 10 = 40.
- From 50-59, the tens digit is always 5. This appears 10 times. The sum is 5 * 10 = 50.
- From 60-69, the tens digit is always 6. This appears 10 times. The sum is 6 * 10 = 60.
- From 70-79, the tens digit is always 7. This appears 10 times. The sum is 7 * 10 = 70.
- From 80-89, the tens digit is always 8. This appears 10 times. The sum is 8 * 10 = 80.
- From 90-99, the tens digit is always 9. This appears 10 times. The sum is 9 * 10 = 90.
- For 100, the tens digit is 0. This appears 1 time. The sum is 0 * 1 = 0.
The sum of the tens digits is 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 + 0 = 450.
Ones Place:
- For each decade (10-19, 20-29, etc.), the ones digits cycle through 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The sum of these digits is 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45.
- This cycle repeats 9 times, from 10-99, so the sum of the ones digits from 10-99 is 45 * 9 = 405.
- For 100, the ones digit is 0.
Hundreds Place:
- For 100, the hundreds digit is 1. This appears 1 time. The sum is 1.
Total Sum:
The total sum of all digits is 450 (tens) + 405 (ones) + 1 (hundreds) = 856.
