# Q. What is the sum of all digits which appear in all the integers from 10 to 100 ?

(a) 855

(b) 856

(c) 910

(d) 911

Correct Answer: (b) 856

#### Question from UPSC Prelims 2023 CSAT

**Explanation : **

## Sum of Digits of Intergers

**Sum of the tens digits:**

From 10 to 19, the tens digit is 1.

From 20 to 29, the tens digit is 2.

From 30 to 39, the tens digit is 3.

…

From 90 to 99, the tens digit is 9.

Each group contains 10 numbers.

Therefore, the sum of the tens digits is:**(1+2+3+4+5+6+7+8+9) x 10 = 45 x 10 = 450**

**Sum of the units digits:**

The units digits from 10 to 100 repeat every ten numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

This sequence repeats 9 times (from 10 to 19, 20 to 29, …, 90 to 99).

Therefore, the sum of the units digits is:**(0+1+2+3+4+5+6+7+8+9) x 9 = 45 x 9 = 405**

### Adding the sums of tens and units digits:

**450 + 405 = 855**

However, we must also consider the number 100:

The tens digit is 10, and the units digit is 0.

The sum of digits in 100 is 1 + 0 + 0 = 1.

Adding this to our previous total:**855 + 1 = 856**

Thus, the sum of all digits which appear in all the integers from 10 to 100 is **856**. The correct answer is **(b) 856**.