To solve the problem of finding the least number of beads Rama could have, let's determine the least common multiple (LCM) of the bead counts for the necklaces she makes: 24, 36, and 40.

The LCM is the smallest number that is a multiple of each of these numbers. To calculate the LCM, we will use the prime factorization method:

1. Prime factorization of 24: 24 = 2^3 \times 3^1
2. Prime factorization of 36: 36 = 2^2 \times 3^2
3. Prime factorization of 40: 40 = 2^3 \times 5^1

Next, we take the highest power of each prime number present in the factorization:

• 2^3 (from 24 and 40)
• 3^2 (from 36)
• 5^1 (from 40)

The LCM is thus calculated as follows:

So, the least number of beads Rama could have, ensuring no beads are left over in any type of necklace, is 360.

Let's verify the answer by dividing 360 by each number:

• 360 \div 24 = 15 (Whole number)
• 360 \div 36 = 10 (Whole number)
• 360 \div 40 = 9 (Whole number)

Since 360 can be divided evenly by 24, 36, and 40, the answer is confirmed. Therefore, 360 is the correct answer.