Shortcut Trick
Check options by subtracting the remainder 8 and testing divisibility by 9 (since 18 and 36 are multiples of 9).
(1) 9946 − 8 = 9938 (Sum 29, not div. by 9)
(2) 9944 − 8 = 9936 (Sum 27, divisible by 9)
(3) 9964 − 8 = 9956 (Sum 29, not div. by 9)
(4) 9984 − 8 = 9976 (Sum 29, not div. by 9)
∴ The correct answer is 9944.
Alternate Method
Given: Divisors = 18, 36, 48; Remainder = 8; Target = Greatest 4-digit number.
Formula Used: Required Number = (k × LCM of divisors) + Remainder
⇒ LCM of 18, 36, and 48 = 144
⇒ Greatest 4-digit number = 9999
⇒ Divide 9999 by 144: 9999 ÷ 144 ⇒ Quotient = 69, Remainder = 63
⇒ Greatest 4-digit number divisible by 144 = 9999 − 63 = 9936
⇒ Required number = 9936 + 8 = 9944
∴ The correct answer is 9944.
Additional Information
LCM and Remainder Theorem
When a number is divided by a, b, and c leaving the same remainder r, the number is of the form [LCM(a, b, c) × k] + r.
Divisibility Rule of 9
A number is divisible by 9 if the sum of its digits is divisible by 9. This is a very useful shortcut for LCM-based problems involving multiples of 9.