Aptitude - Problems on H.C.F and L.C.M - Discussion

3:4:5 was arrived at, by cancelling the greatest common factor (How? say we have 3 no.s 100,200,300. To write them in the ratio form we would cancel the greatest common factor from all these three no.s which is 100, to give us the ratio - 1:2:3).

Therefore, the numbers we are looking for in the question, are- hcf*3, hcf*4, hcf*5.
We are given that the LCM of these numbers is 2400, i.e LCM(hcf*3, hcf*4, hcf*5) = 2400
So,
n1 = hcf * 3,
n2 = hcf * 4,
n3 = hcf * 5,

Therefore, LCM = hcf*3*4*5 (LCM is the least no. that IS DIVIDED COMPLETELY by all three numbers).
thus,
hcf *3*4*5 = 2400.
hcf = 40.

Then the numbers are 120, 160 ,200 [ratio 3:4:5 holds, LCM - 2400 holds].

In the explanation answer,the formula has been used is
Factor of number 3x={3,x}
Factor of number 4x={4,x}
Factor of number 5x={5,x}
Therefore,LCM is a product of {3,4,5,x} [x has taken once as it is common] i.e.LCM=60x
60x=2400
x=40
But,
If we take another formula that
Product of all numbers = Product of LCM and HCF
3x*4x*5x=2400*x
60x3[x to the power 3 or x cube]=2400x
x3[x to the power 3 or x cube] 2400
------------------------------ = ------
x 60
x2[x to the power 2 or x square]=40

therefore, x= square root of 40
then?
Can anybody solve my query???

I did like this :

Once I found the x = 40.

I understood that the three numbers are 120, 160, 200

I applied the formula like this :

Product of the numbers = lcm of the numbers * G.C.F of the number.

So G.C.F = product\L.C.M.
So G.C.F = (120*160*200)\2400 =1600.

Which is not the correct answer. Since the concept of G.C.F *L.C.M = product is applicable only for two numbers and not more than that.

Its wrong. LCM * HCF = product of numbers is only valid for 2 numbers.

Consider an example for no 2, 15, 20;

LCM = 2*2*3*5 = 60.

While HCF of these two numbers is 1.

Now as per your logic LCM*HCF = product of 2 numbers.

Hence as per this:

60*1 = 2*15*20.

60 != 600.

Hence your logic is invalid please help regarding that.

Yes @Arunkmar S.

@Kishore.

Here the proportion(3:4:5) can be written like this 3x, 4x, & 5x.
So LCM (3x,4x,5x)= x*3*4*5 = 60x.
But the LCM is 2400(given) so, your x=2400/60 = 40.
The required numbers are 3*40=120, 4*40=160, 5 * 40= 200.
Now find the HCF of 120, 160, 200 i.e. 40.

Simple right?

Given the numbers are in ratio 3:4:5 so the actual numbers will be 3k,4k,5k.
Multiplication of all the numbers = h.c.f*l.cm.
H.c.f will be k and given l.c.m is 2400,
Therefore,3k * 4k * 5k = k * 2400,
By solving this we will get k = 40.

Let number be 3a, 4a and 5a.

3 = 3 x 1
4 = 2 x 2
5 = 5 x 1.

LCM(3,4,5) = 3 x 2^2 * 5 = 60.

60a = 2400,
a= 40.

Now, Numbers are 120, 160, 200.
120 = 2^3 * 5 * 3.
160 = 2^5 * 5,
200 = 2 ^3 * 5^2.

HCF(120,160,200) = 2^3* 5^2 = 200.

Can someone explain these?

Factors of 4 : 1,2,4
Factors of 40 : 1,2,4,5,8,10,20,40

Therefore HCF is 4.

But we should only take pairs if 3,40 are co-primes means their HCF should be 1.
Here HCF should be 1 but it is 4.

600,800,1000 commom divisor is :200

600)800(1
600
---------
200)600(3
600
---------------
0
---------

200)1000(5
1000
-------
0
----------


so 200 is the H C F.............

Given the ratio is 3:4:5.
Consider the HCF of the three numbers is x i.e., is 3x, 4x, 5x.
LCM of 3x, 4x, 5x=60x.
Given that LCM of the three numbers is 2400.
Therefore 60x=2400.
x=40.
So the HCF value is 40.