Question

Que.24, Find the two numbers nearest to the smallest 5-digit number which are exactly divisible by each of 2, 3, 4, 5, 6 and 7?​

Answer 1

Answer:

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Answer 2

Lets see how we get the answer.

First of all check that if any number is divisible by 6, then it is divisible by 2 and 3 as 2 and 3 are factors of 3.

So, the question reduces to smallest number of 5 digit which is exactly divisible by 4,5,6,7.

To find the number,we have to find the common multiple of 4,5,6,7

.Taking the LCM of 4,5,6,7, we get

12 * 5 * 7 =420.

So, we have to find the smallest number of 5 digits which is exactly divisible by 420.

Lets multiply the number by 20, we get

420*12 = 8400.

Now , we are very close.

Now, we have to add some number X to 8400 such that result becomes a 5 digit number and also X is divisible by 420.

Adding 420 * 4( =1680) to 8400, we get

8400 + 1680 =10080 which is the smallest number.

Hence , 10080 is the required answer.

Note that, we have to add a number greater than 1600 to 8400 to make it a 5 digit number.The smallest multiple of 420 greater than 1600 is 1680.

So , we add it to 8400.

Property Used -

According to Euclid's theorem, if A and B are divisible by a number C, then

A + B and A - B are also divisible by C.

Here, we have used this to get the answer.

8400 + 1680 =10080.

Hope that helps. :)

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