Shortcut Trick
Find the LCM of the given numbers 21, 35, and 56, which is 840.
The smallest five-digit number is 10000.
Divide 10000 by 840 to find the remainder: 10000 ÷ 840 ⇒ Remainder = 760.
The least multiple of 840 above 10000 is 10000 + (840 − 760) = 10080.
∴ The correct answer is 10080.
Alternate Method
Given: Divisors = 21, 35, and 56. Target = Least five-digit number.
Formula Used: Required number = Smallest n-digit number + (LCM − Remainder of Smallest n-digit number / LCM).
⇒ Prime factorization of 21 = 3 × 7
⇒ Prime factorization of 35 = 5 × 7
⇒ Prime factorization of 56 = 2 × 2 × 2 × 7 = 23 × 7
⇒ LCM = 23 × 3 × 5 × 7 = 8 × 3 × 5 × 7 = 840
⇒ Smallest five-digit number = 10000
⇒ 10000 ÷ 840 ⇒ Quotient = 11, Remainder = 760
⇒ Required number = 10000 + (840 − 760)
⇒ Required number = 10000 + 80 = 10080
∴ The correct answer is 10080.
Additional Information
Least Common Multiple (LCM)
The smallest positive integer that is divisible by each of the given numbers. It is found by taking the highest power of all prime factors involved.
Greatest n-digit Number Divisible by X
To find the greatest n-digit number divisible by a set of numbers, divide the largest n-digit number (e.g., 99999) by their LCM and subtract the remainder.
Divisibility Rule for 8
A number is divisible by 8 if its last three digits are divisible by 8. For 10080, 080 is divisible by 8.