What least number must be added to 1023, so that the sum is completely…

To find the least number that must be added to 1023 so that the sum is completely divisible by 23, we can follow a systematic approach using division and remainders.

Detailed Step-by-Step Solution
Perform the Initial Division:
First, divide 1023 by 23 to determine how far the number is from the nearest multiple of 23.

1023 / 23 = 44 with a remainder.

To find the remainder, calculate 23 multiplied by 44, which equals 1012.

Now, subtract this from the original number: 1023 - 1012 = 11.

The remainder is 11. This means 1023 is 11 units greater than the multiple 1012.

Determine the Required Addition:
Since we want the next multiple of 23, we need to find how many more units are required to reach the next divisor.

The current remainder is 11, and the divisor is 23.

To reach the next multiple, calculate the difference between the divisor and the current remainder: 23 - 11 = 12.

Therefore, by adding 12 to 1023, we will reach the next multiple of 23.

Verify the Result:

1023 + 12 = 1035.

Dividing 1035 by 23: 1035 / 23 = 45.

Since 45 is a whole number, 1035 is perfectly divisible by 23, confirming that 12 is the correct number to add.