To find the LCM use prime factorization:
15: 5 x 3
25: 5 x 5
40: 5 x 8 = 5 x 2 x 2 x 2
75: 15 x 5 = 5 x 5 x 3
LCM: Every prime factor such that none are repeated:
5 x 3 (from 15) x 5 (from 25) x 2 x 2 x 2 (from 40) and since two 5s and 3 are already there you dont take anything from 75.
so LCM is 5 x 3 x 5 x 2 x 2 x 2 = 600

Then just find the greatest 4 digit number divisible by 600. Which is shown and explained well by the problem explanation.

Pavan said:
(Mon, Dec 13, 2010 02:01:49 PM)

Why subtracted 9999-399?

Vishnu said:
(Fri, Jan 28, 2011 01:46:33 AM)

Just check if the answers are divisible by 40(one of the numbers given)...It's easy to notice that 9000,9400 and 9800 will not be divisible. So, it's obviously 9600.

Aparna said:
(Fri, Jan 28, 2011 03:00:50 AM)

@vishnu.

How 9000, 9400 and 9800 are not divsible by 40?.

It is divisible by 40.

Gwene said:
(Sun, Jan 30, 2011 07:32:24 AM)

Why only 9999? among those 4 numbers 9800 is greater so why not 9800?

Gwene said:
(Sun, Jan 30, 2011 07:35:37 AM)

Vishnu why should we divide by 40?

Mehar said:
(Mon, Jan 31, 2011 02:16:49 AM)

hai vishnu...why should divide with only 40 why cant we do that with reaming no given there?

Jyoti said:
(Mon, Jan 31, 2011 03:03:38 AM)

@vishnu.

Their is no need to divide by 40.

Debashree said:
(Mon, Feb 14, 2011 09:21:16 PM)

Wow should we divide 9999 by 600

i) 1st we find the greatest 4 digit number

ii) Then find the lcm of 15,25,40 ,75 ,is 600 ,so 600 is least number by which 15,25,40,75.so all the multiple of 600 is divisible by 15,25,40,75

iii) Then find multiple of 600 which is less than 999

iv) So we divide 9999 by 600

and get 399 as remainder,

If substract 399 from 9999 tnen we get that number

We have to find greatest 4 digit number so it may be between 9000 to 9999 let the number be x

this x should be divisible by 15, 25, 40 and 75 so take common least nnumber from all factors i.e l.c.m=600

therefore x is also divisible by 600 that means when x is divided by 600 remainder should be '0' but how can we find that number??

that number x lies between 9000 to 9999

take any number between them i.e take 9000,..9010,...9090...9900...9997,9998,9999 as your wish

so insted of 9999 here iam taking 9998 which is divided by 600

so remainder is 398 so substract 398 from 9998 i.e 9998-398=9600

why we wre substracting is because x should be divisible 600 and
remaider should be '0' so x is 9600 thats it

Bhargav said:
(Sun, May 29, 2011 01:39:37 PM)

Good answer rahul.

Neha said:
(Thu, Jun 30, 2011 03:10:20 AM)

Find the LCM of 15, 25, 40 and 75 which is 600.

Now, find out the multiples of 600 i.e.
600*1=600
600*2=1200
600*3=1800
........
600*16=9600
600*17=10200

We can see that the greatest number of four digits is 9600 and hence 9600 is the answer.

Rachna said:
(Thu, Jun 30, 2011 04:21:55 AM)

Greatest number of 4-digits is 9999.

L.C.M. of 15, 25, 40 and 75 is 600.
(I understood till above, but how come 9999 divided by 600 is 399??)

On dividing 9999 by 600, the remainder is 399??

Required number (9999 - 399) = 9600??

Sameer said:
(Sun, Jul 10, 2011 02:46:11 PM)

Find LCM AND HCF OF
x3+2x2-4x-8 and2x3+7x2+4x-4

Arunav said:
(Tue, Jul 26, 2011 03:18:32 PM)

There can be another approach. Just find the L.C.M. of the nos. The L.C.M. comes to be 600. Now check which is the greatest no. which is getting divided by 600.

Numbers of option B and option D are not getting divided by 600 and between option B and C (which are getting divided by 600) the number of option C is the greatest no. divisible by 600

In this way I guess it would be quicker than dividing 9999 and then subtracting the remainder.

Lakhan said:
(Fri, Dec 30, 2011 04:41:21 PM)

Friends will you please explain how the LCM came 600 in detail ?

Lakhan said:
(Fri, Dec 30, 2011 04:45:28 PM)

@Rahul

How can you ignore 75= 15 x 5 = 5 x 5 x 3 while calculating LCM ?

Sam said:
(Tue, Jan 3, 2012 04:05:32 PM)

Great explanation Debashree.

Pk Rojer said:
(Mon, Jan 9, 2012 03:00:56 PM)

Instead of calculating LCM Why we don't go for calculating HCF.

Dr.Deepshikha Saini said:
(Fri, Mar 16, 2012 08:11:28 PM)

Debashree you are superb. Thanks for such a wonderful explanation.

Akshata said:
(Sat, Apr 14, 2012 03:25:32 PM)

@Arunav.

Your explanation seems to be quick and fast. Can you elaborate and let me know who will it work.

My funds of finding LCM in this case 600 is clear.

But when you are writing "Numbers of option B and option D are not getting divided by 600 and between option B and C (which are getting divided by 600) the number of option C is the greatest no. Divisible by 600" is not clear. Just let me know the explanation.

In this way I guess it would be quicker than dividing 9999 and then subtracting the remainder.

Aslam said:
(Tue, Apr 17, 2012 08:51:29 PM)

Please calculate the remainder.

Alwin said:
(Wed, May 9, 2012 08:32:10 AM)

To find LCM
15=3*5
25=5*5
40=2*2*2*5
75=3*5*5
Every number written in the form of multiples of prime number
Now we will write as
15=> 3 5
25=> - 5 5
40=> - 5 - 2 2 2
75=> 3 5 5 - - -
___________________
3 5 5 2 2 2
Then
3*5*5*2*2*2=600

Hence the result

Dileep Kumar M B said:
(Wed, Aug 29, 2012 04:50:35 PM)

How a person could know whether to take LCM or HCF? can anybody explain me when should we use these LCM and HCF ?

Rahul Vasu said:
(Sun, Nov 25, 2012 10:48:21 PM)

Calculating without LCM method

Shortcut way to get 600

In the option A) is given 9000 and B) 9400 and C) 9600 and D) 9800

To subtract from highest number first i.e (D-C) 9800-9600= 200 and (B-A) 9400-9000= 400

Now (D-C)=200
(B-A)=400

Add 200 + 400= 600

The questions says greatest four digit that means in every first digit is 9..so therefore there will be 4 greatest digits of 9 i.e is 9999

Hence 9999-600= 9,399
(Therefore the first digit is 9 so the rest 3 digit is 399 and hence 9999 - 399)

9999-399= 9600

Ravi said:
(Sun, Dec 2, 2012 03:11:03 PM)

We need to find highest 4 digit least common multiplier. So lcm of given numbers is 600. It's a three digit number but need to find 4 digit number. 600*8=4800 but I still need to be in range below 10000. 4800*2=9600.

Ram said:
(Mon, Jun 10, 2013 10:50:49 PM)

How to find 600 is LCM?

Priya said:
(Sun, Jul 28, 2013 12:47:00 PM)

LCM of 15: 5 X 3.

25: 5^2.

40: 5 X 2^3.

75: 5^2 X 3.

From each highest power of 5 = 5^2, 2 = 2^3, 3 = 3.

So 5^2 X 2^3 X 3 = 600.

Vinod said:
(Tue, Aug 27, 2013 01:59:21 PM)

I don't know how to do LCM & HCF, anyone can you help me how to do?

S. Nganba Ypk said:
(Sat, Aug 31, 2013 02:19:41 PM)

L.C.M. of 15, 25, 40 and 75 is 600.

Now, We find out the multiples of 600 of lowest/smallest five digit number i.e.600*17=10200.

Therefore, Greatest four digit number divisible by 15, 25, 40 and 75 can easily found by 600*16 i.e 9600.

This is simplest & easiest method.

Ankit said:
(Fri, Sep 13, 2013 12:58:25 AM)

When greatest FOUR Digit number is to be found it means the number wont exceed 9999 so when we take 600 modulus 9999 we get remainder 399 which means that we are having 399 left after dividing 9999 with 6000 so we subtract 399 from 9999 so that remainder becomes zero and we get the greatest 4 digit number.

As the LCM of four number is 600 the number divisible by them are multiple of 600 i.e.

600
1200
1800
2400
3000
3600
4200
4800
5400
6000
6600
7200
7800
8400
9000
9600 <----- this is the greatest 4 digit no. divisible
10200

Please tell why we have to take lcm instead instead of hcf and in which situation I have to take lcm or hcf?

Tejas Kumbhar said:
(Sat, Feb 8, 2014 04:13:28 PM)

The Logic here is this:

We have to find the greatest 4 digit number which is divisible by 15, 25, 40 and 75, hence it is a multiple of all these.

The LCM(15, 25, 40, 75) = 600 i.e. the least common multiple.

Hence all other common multiples of 15, 25, 40 and 75 will be multiples of 600 as well.

So to find the highest 4 digit one among all the common multiples, divide all the options by 600 to check which is perfectly divisible by 600. We get 9600, which is the answer.

Shubhangi said:
(Mon, Apr 14, 2014 11:53:03 AM)

Why do we take LCM of the given numbers, but not HCF?