Shortcut Trick
The required number is exactly divisible by 18, so it must also be divisible by 9.
By the divisibility rule of 9, the sum of its digits must be a multiple of 9.
From the given options (8, 6, 9, 12), only 9 is a multiple of 9.
Alternatively, the smallest 4-digit multiple of their LCM (360) is 360 × 3 = 1080.
Sum of digits of 1080 = 1 + 0 + 8 + 0 = 9.
∴ The correct answer is 9.
Alternate Method
Given:
Divisors: 8, 10, 12, 18, and 24
Formula Used:
Required number = Smallest 4-digit number + (LCM − Remainder)
Calculations:
⇒ Prime factorization of given numbers:
⇒ 8 = 23
⇒ 10 = 2 × 5
⇒ 12 = 22 × 3
⇒ 18 = 2 × 32
⇒ 24 = 23 × 3
⇒ LCM = 23 × 32 × 5 = 8 × 9 × 5 = 360
⇒ The smallest 4-digit number is 1000.
⇒ 1000 ÷ 360 gives a quotient of 2 and a remainder of 280.
⇒ To make it perfectly divisible by 360, we add the difference between the divisor and the remainder.
⇒ Required Number = 1000 + (360 − 280) = 1080
⇒ Sum of the digits = 1 + 0 + 8 + 0 = 9
∴ The correct answer is 9.
Additional Information
LCM and HCF Relation
The product of two numbers is equal to the product of their LCM and HCF: a × b = LCM × HCF.
Greatest n-digit number divisible by given numbers
To find the greatest n-digit number divisible by given numbers, find their LCM, divide the greatest n-digit number by the LCM, and subtract the remainder from the greatest n-digit number.
Divisibility Rule of 9
A number is perfectly divisible by 9 if the sum of its digits is a multiple of 9. This rule is highly useful for cross-verifying answers involving multiples of 9 or 18.