The least number which should be added to 11557 so that the sum is exactly divisible by 3, 4, 5, 6, 8, and 10 is:

Given:

The number to be added to 11557 so that the sum is divisible by 3, 4, 5, 6, 8, and 10.

Concept Used:

We need to find the Least Common Multiple (LCM) of the numbers 3, 4, 5, 6, 8, and 10. Then, we need to calculate how much more should be added to 11557 to make the sum divisible by this LCM.

Calculation:

Find the LCM of 3, 4, 5, 6, 8, and 10:

Prime factorization of each number:

3 = 3

4 = 2²

5 = 5

6 = 2 × 3

8 = 2³

10 = 2 × 5

The LCM will be the product of the highest powers of all prime factors:

LCM = 2³ × 3 × 5 = 120

Now, we need to find the least number to be added to 11557 so that the sum is divisible by 120.

Divide 11557 by 120:

⇒ 11557 ÷ 120 = 96 remainder 37

The remainder is 37. Therefore, we need to add (120 - 37) = 83 to 11557 to make the sum divisible by 120.

∴ The least number that should be added to 11557 is 83.