Question

Find the greatest number of 5 digits exactly divisible by 12, 15 and 36. Find the smallest number which when increased by 20 is exactly divisible by 90 and 144 0. Find the smallest.

Answer 1

Hint: There are two parts of the questions. In the first part of the question, we have to find the greatest number of 5 digits which is exactly divisible by 12, 15 and 36. For that, we will find the L.C.M of these numbers. And then we will divide the greatest 5 digit number which is 99999 by the L.C.M obtained. We will subtract the remainder from the number 99999. Obtained numbers will be the greatest number of 5 digits which is divisible by 12, 15 and 36.
In the second part of the question, we have to find the smallest number which when increased by 20 is exactly divisible by 90 and 144. For that, we will find the L.C.M of 90 and 144. Then we will subtract 20 from the L.C.M to get the required number.

Complete step-by-step answer:
In the first part of the question, we have to find the greatest number of 5 digits exactly divisible by 12, 15 and 36. For that, we will find the L.C.M of 12, 15 and 36.
We know greatest 5-digit number $=99999$
First we will find all the factors of 12, 15 and 36.
$\begin{align}
  & 12=2\times 2\times 3 \\
 & 15=3\times 5 \\
 & 36=3\times 3\times 2\times 2 \\
\end{align}$
The L.C.M of 12, 15 and 36 $=2\times 2\times 3\times 3\times 5=180$
Now, we will divide 99999 by 180 and we will find its remainder.

The remainder, which we have got after division of 99999 by 180, is 99.
Now, we will subtract the remainder from 99999.
$99999-99=99900$
Thus, the required number is 99900.
In the second part of the question, we have to find the smallest number which when increased by 20 is exactly divisible by 90 and 144.
First, we will find the L.C.M of 90 and 144.
We will find the factors of the 90 and 144.
$\begin{align}
  & 90=2\times 3\times 3\times 5 \\
 & 144=2\times 2\times 2\times 2\times 3\times 3 \\
\end{align}$
L.C.M of 90 and 144 $=2\times 2\times 2\times 2\times 3\times 3\times 5=720$
To get the required number, we will subtract 20 from 720.
So the required number is $=720-20=700$
Thus, the smallest number, which becomes exactly divisible by 90 and 144 after adding 20 to it $=700$

Note: Since we have calculated the L.C.M here, we need to know that term properly. L.C.M is a method to find the smallest common multiple of two or more numbers.
Some important properties of L.C.M are:-
L.C.M of any two or more numbers can’t be less than any of the numbers.
L.C.M of the co-prime numbers is equal to the product of the numbers, where co-prime number is the set of numbers which have only one as their common factor.
L.C.M of fractions is equal to the ratio of the L.C.M of the numerator to the H.C.F of the denominator.