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.