Question

determine the two numbers nearest to 10000which are exactly divisible by each of 2,3,4,5,6,and 7​

Answer 1

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:

2∣∣234567−−−−−−−−−−−−−−−−3∣∣132537−−−−−−−−−−−−−−−−∣∣112517−−−−−−−−−−−−−−−−

.

So, we have got LCM (Least common multiple) of 2, 3, 4, 5, 6 and 7 as 2×3×1×1×2×5×1×7=420.

The LCM (Least common multiple) of 2, 3, 4, 5, 6 and 7 is 420.

Let us divide 10000 with the obtained LCM.

420)10000(23 840 −−−− 1600 1260 −−−−− 340

.

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 (10000−340)=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 (9660+420)=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.