**1. Find the greatest 6- digit number among the following which is exactly divisible by 24, 15 and 36.**

(A) 999999

(B) 999720

(C) 999750

(D) 999820

**Ans: (B) 999720**

**Solution: ** Find the Greatest 6-Digit Number Divisible by 24, 15, and 36

To find the greatest 6-digit number among the given options that is exactly divisible by 24, 15, and 36, we first need to determine the Least Common Multiple (LCM) of these numbers.

**Step 1: Prime Factorization**

\[

24 = 2^3 \times 3

\]

\[

15 = 3 \times 5

\]

\[

36 = 2^2 \times 3^2

\]

**Step 2: Find the LCM**

LCM is found by taking the highest powers of all prime factors:

\[

2^3 \quad (\text{from } 24)

\]

\[

3^2 \quad (\text{from } 36)

\]

\[

5 \quad (\text{from } 15)

\]

\[

\text{Therefore, LCM} = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360

\]

**Step 3: Check the Options**

\[

\frac{999999}{360} = 2777.775

\]

This is not an integer, so 999999 is not divisible by 360.

\[

\frac{999720}{360} = 2777

\]

This is an integer, so 999720 is divisible by 360.

\[

\frac{999750}{360} = 2777.0833

\]

This is not an integer, so 999750 is not divisible by 360.

\[

\frac{999820}{360} = 2777.2777

\]

This is not an integer, so 999820 is not divisible by 360.