Introduction:
To find the least number that is divisible by 10, 20, 30, 40, and 45, we need to find their least common multiple (LCM). The LCM is the smallest number that is divisible by all the given numbers.

Step 1: Prime Factorization:
To find the LCM, we start by prime factorizing each of the given numbers.

- 10 = 2 * 5
- 20 = 2^2 * 5
- 30 = 2 * 3 * 5
- 40 = 2^3 * 5
- 45 = 3^2 * 5

Step 2: Identify Common Prime Factors:
Next, we identify the common prime factors among the prime factorizations of the given numbers. In this case, the common prime factors are 2, 3, and 5.

Step 3: Maximum Exponent:
We determine the maximum exponent for each common prime factor. In this case, the maximum exponent for 2 is 3, for 3 is 2, and for 5 is 1.

Step 4: Calculate LCM:
To find the LCM, we multiply the common prime factors raised to their respective maximum exponents.

LCM = 2^3 * 3^2 * 5^1
= 8 * 9 * 5
= 360

Step 5: Add 10:
To find the least number which on adding 10 is exactly divisible by 10, 20, 30, 40, and 45, we add 10 to the LCM.

Least Number = LCM + 10
= 360 + 10
= 370

Explanation:
The least common multiple (LCM) of 10, 20, 30, 40, and 45 is 360. Therefore, the least number which on adding 10 is exactly divisible by these numbers is 370.