Question

Find the largest number of four digits exactly divisible by 12, 15, 18 and 27.
A.5678
B.2784
C.2793
D.9720

Answer 1

Hint: - LCM is elaborated as Least Common Multiple, which is the lowest common factor among the integers. To find the LCM of the given numbers (Lowest common number), which is divisible by all numbers, the method includes basic factorization of the numbers to find factors that are multiplied together to form a number.

Complete step-by-step answer:
In this question, to find the largest number which will be exactly divisible by the given numbers, find their LCM by finding the factors and divide the LCM with the largest number 9999 to check whether they are divisible and if not find the difference.
Given the numbers are 12, 15, 18 and 27
The largest 4 digit number known is 9999
For largest number find the LCM by factorizing of the numbers
\[
   2\underline {\left| {12,15,18,27} \right.} \\
   3\underline {\left| {6,15,9,27} \right.} \\
   3\underline {\left| {2,5,3,9} \right.} \\
   2,5,1,3 \\
 \]
Hence the factors of each number are:
\[
  12 = 2 \times 3 \times 2 \\
  15 = 3 \times 5 \\
  18 = 2 \times 3 \times 3 \\
  27 = 3 \times 3 \times 3 \\
 \]
So the LCM of the three numbers will be:
\[LCM\left( {12,15,18,27} \right) = 2 \times 2 \times 3 \times 3 \times 3 \times 5 = 540\]
Hence the LCM of numbers \[12,15,18,27\] is\[540\]
Now check whether the 4 digit greatest number is divisible by 540
On dividing\[\dfrac{{9999}}{{540}} = 18\dfrac{{279}}{{540}}\], 279 is the remainder; hence we can say 9999 is not exactly divisible by 540, hence subtract
\[\left( {9999 - 279} \right) = 9720\]
The number is exactly divisible by 540 hence 9720 is the largest 4 digit number divisible by 12, 15, 18 and 27.

Note: LCM of given numbers is exactly divisible by each of the numbers. During the LCM calculation, students must know the tables of various numbers, and they have to perform the operations step by step. Lastly, they have to multiply all the numbers by which they are dividing the given set of numbers.