Question

Find the two numbers nearest to the $10000$ which are exactly divisible by each $2,3,4,5,6and7$.

Answer 1

The numbers which are exactly divisible by $2,3,4,5,6and7$ are the multiples of the LCM of the given numbers.
∴ $\mathrm{LCM}=2\times 2\times 3\times 5\times 7=420$
Now, dividing $10000\mathrm{by}420$

We get remainder $=340$
Step 1:

∴ Number just less than $10000$ and exactly divisible by the given numbers $=10000-340$

$=9660$

Step 2:

Number just greater than $10000$ and exactly divisible by the given numbers $=10000+\left(420-340\right)$

$=10080$

Hence $9660\mathrm{and}10080$ are the two numbers nearest to the $10000$ which are exactly divisible by each $2,3,4,5,6and7$.