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

The smallest number which is exactly divisible by \(2, 3, 4, 5, 6,\) and \(7\) is their L.C.M.
But, we have to find two numbers nearest to \(10000\) which are exactly divisible by the given numbers, i.e. \(2, 3, 4, 5, 6\) and \(7.\) Clearly, such numbers are multiples of the L.C.M. of the given numbers.

To find the L.C.M. of \(2, 3, 4, 5, 6,\) and \(7,\) we have,

\(\therefore L.C.M.=2\times 3\times 2\times 5\times 7=420\)
The number nearest to \(10000\) and exactly divisible by each of \(2, 3, 4, 5, 6, \)and \(7\) should also be exactly divisible by their L.C.M. i.e., \(420. \)
Let us now divide \(10000\) by \(420.\)

We find that the remainder is \(340.\)
Number just less than \(10000\) and exactly divisible by \(420\) is \(10000-340=9660\)
Number just greater than \(10000\) and exactly divisible by \(420\) is \(=10000+(420-340)=10080\)
Hence, the required numbers are \(9660\) and \(10080.\).

The two numbers nearest to 10000 which are exactly divisible by each of 2, 3, 4, 5, 6 and 7, are _____.(A) 9660, 10080(B) 9320, 10080(C) 9660, 10060

(D) 10340, 10080