Explanation:

To find the least number that must be added to 1056 so that the sum is completely divisible by 23, we need to find the remainder when 1056 is divided by 23.

Step 1: Finding the remainder
To find the remainder, we divide 1056 by 23 using long division method.

45
23) 1056
- 92
-----
160
- 138
-----
220
- 207
-----
130
- 115
-----
150
- 138
-----
120
- 115
-----
50
- 46
-----
40
- 23
-----
17

The remainder when 1056 is divided by 23 is 17.

Step 2: Finding the number to be added
To make the sum completely divisible by 23, we need to add the difference between 23 and the remainder (17).

23 - 17 = 6

So, the least number that must be added to 1056 to make the sum completely divisible by 23 is 6.

Step 3: Checking the options
Now, let's check the given options to identify the least number.

a) 2
When 2 is added to 1056, the sum becomes 1058. However, 1058 is not divisible by 23.

b) 3
When 3 is added to 1056, the sum becomes 1059. However, 1059 is not divisible by 23.

c) 18
When 18 is added to 1056, the sum becomes 1074. 1074 is divisible by 23.

d) 21
When 21 is added to 1056, the sum becomes 1077. 1077 is divisible by 23.

Out of the given options, the least number that makes the sum completely divisible by 23 is 18 (option C). However, the correct answer mentioned is option A, which is incorrect.

Therefore, the correct answer should be option C) 18.