Question

Find the two numbers nearest to 10000 which are exactly divisible by each of 2, 3, 4, 5, 6 and 7?
(a) 9660, 10080
(b) 9320, 10080
(c) 9660, 10060
(d) 10340, 10080

Answer 1

Hint: We start solving the problem by recalling the fact that if a number has to be divisible by a group of numbers, then the given number has to be divisible by the LCM (Least common multiple) of the group of numbers. So, we find the LCM of the given numbers 2, 3, 4, 5, 6 and 7 by which we divide the given number 10000 to get the remainder. We then subtract the obtained remainder from 10000 to get one of the nearest numbers to 10000 that is divisible by 2, 3, 4, 5, 6 and 7. We then add LCM to the nearest number obtained to get the next nearest number to 10000 divisible by 2, 3, 4, 5, 6 and 7.

Complete step-by-step answer:
According to the problem, we have to find the two nearest numbers to 10000 that are exactly divisible by each of the numbers 2, 3, 4, 5, 6 and 7.
We know that if a number has to be divisible by a group of numbers, then the given number has to be divisible by the LCM (Least common multiple) of the group of numbers.
So, let us find the LCM (Least common multiple) of 2, 3, 4, 5, 6 and 7.
The LCM is calculated as follows:
\[\begin{align}
  & 2\left| \!{\underline {\,
  \begin{matrix}
   2 & 3 & 4 & 5 & 6 & 7 \\
\end{matrix} \,}} \right. \\
 & 3\left| \!{\underline {\,
  \begin{matrix}
   1 & 3 & 2 & 5 & 3 & 7 \\
\end{matrix} \,}} \right. \\
 & \left| \!{\underline {\,
  \begin{matrix}
   1 & 1 & 2 & 5 & 1 & 7 \\
\end{matrix} \,}} \right. \\
\end{align}\].
So, we have got LCM (Least common multiple) of 2, 3, 4, 5, 6 and 7 as $2\times 3\times 1\times 1\times 2\times 5\times 1\times 7=420$.
The LCM (Least common multiple) of 2, 3, 4, 5, 6 and 7 is 420.
Let us divide 10000 with the obtained LCM.
\[\begin{align}
  & \left. 420 \right)10000\left( 2 \right.3 \\
 & \underline{\text{ 840 }} \\
 & \text{ 1600} \\
 & \underline{\text{ 1260 }} \\
 & \text{ 340} \\
\end{align}\].
On dividing 10000 with 420, we get the remainder 340. This means that 10000 is 340 more than the number that is exactly divisible by 420. So, we subtract 340 from 10000 in order to get the nearest number that is exactly divisible by 420.
The nearest number to and less than 10000 which is divisible by 420 is $\left( 10000-340 \right)=9660$.
Now, we need to find the nearest number to and greater than 10000 that is exactly divisible by 420.
We add 420 to 9660 in order to get the next number that is divisible by 420.
So, we have the next number as $\left( 9660+420 \right)=10080$.
We can see that 10080 is greater than 10000 and divisible by 420. So, this is the next nearest number to 10000 that is divisible by 420.
We have found the two numbers nearest to 10000 which are exactly divisible by each of 2, 3, 4, 5, 6 and 7 as 9660 and 10080.
∴ The two numbers nearest to 10000 which are exactly divisible by each of 2, 3, 4, 5, 6 and 7 are 9660 and 10080.
The correct option for the given problem is (a).

Note: We can also subtract 9660 with 420 to get the number that is divisible by 2, 3, 4, 5, 6 and 7 but it will not be as near when compared to 10080. Whenever we get this type of problem, it is better to solve by finding the LCM (Least common multiple) as it will take more time to divide with each and every variable and finding the exact number. Similarly, we can expect the problems to find every number that is divisible by 2, 3, 4, 5, 6 and 7 less than a particular number.