Sure, let's solve this step-by-step.
1. Find the Least Common Multiple (LCM) of 40, 60, and 100:
- The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set.
Let's denote the LCM of 40, 60, and 100 as [tex]\( \text{LCM}(40, 60, 100) \)[/tex].
After calculating, we find:
[tex]\( \text{LCM}(40, 60, 100) = 600 \)[/tex]
2. Find the necessary value to be added to 15:
- We need to find the least number [tex]\( x \)[/tex] such that when added to 15, the result is divisible by the LCM of 40, 60, and 100.
- Mathematically, we need [tex]\( 15 + x \)[/tex] to be divisible by [tex]\( 600 \)[/tex].
3. Perform the modulo operation to find the value of [tex]\( x \)[/tex]:
- The expression [tex]\( (15 + x) \% 600 \)[/tex] should equal 0, which implies that [tex]\( x \)[/tex] should be chosen such that [tex]\( 15 + x \)[/tex] is a multiple of [tex]\( 600 \)[/tex].
- To determine [tex]\( x \)[/tex], we do the following common remainder calculation:
[tex]\[
15 \% 600 = 15
\][/tex]
Since [tex]\( 15 \)[/tex] is the remainder, the value [tex]\( x \)[/tex] we need to add to [tex]\( 15 \)[/tex] to make it divisible by [tex]\( 600 \)[/tex] is:
[tex]\[
x = 600 - 15 = 585
\][/tex]
Therefore, the least number to which 15 may be added so that the sum is exactly divisible by 40, 60, and 100 is [tex]\( \boxed{585} \)[/tex].
