HCF and LCM problem

radha gogia

Please tell the approach of solving ques 18 and 19 ,I cant understand the logic out here.

18. Least number that will be divided by 3,4,6,8,10,11 would be LCM of these numbers.
Factorize the numbers to get the LCM.
3 => 3*1
4 => 2*2
6 => 3*2
8 => 2*2*2
10 => 5*2
11 => 11*1

so LCM = 3 * 2 * 2 * 2 * 5 * 11.
hence the least perfect square which is divisible by the numbers 3,4,6,8,10,11 is LCM* some number which makes it a perfect square.

So multiply the above LCM with prime factors to make it a perfect square.
We have one 3, to make it a perfect square, multiply another 3.
similarly, we have three 2's, to make it a perfect square, we just need 1 more 2.
proceed with same logic for 5 and 11.

Hence the ans is 2*2*2*2 * 3*3 * 5*5 * 11*11.

19.
Greatest 4 digit number is 9999.
Now let us find the lcm of 10,11,15,22
By primary methods, the lcm is 110.

The greatest 4 digit number which is divisible by 10,11,15,22 is also divisible by the LCM.

Hence, lets find the greatest 4 digit number divisible by 110.
This number turns out to be 9900.

Observe the relation between divisors and the remainders, they are in form of
R=D-7 where R is the remainder and D is the divisor.

Thus the required number is 9900-7 = 9893.