Question

find the least number of 5 digits exactly divisible by each of 3,4,5,6,7 and 8

Answer 1

is the answer. 10004.40

because, the LCM of all given numbers 3,4,5,6,7,8 is 840

but we want smallest 5 digit number, so, the smallest 5 digit number that is divisible by all given number must be close to 10000 (which is smallest 5 digit number of all)

so divide the 10000 by 840
11.9047....

so, it is approximately equal to 11.91

so 840 x 11.91 = 10004.4

Hence
so is the smallest five digit number that is divisible by all given numbers.

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Answer 2

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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