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

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.

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.

2400/3 * 4 * 5 = 40.

Let's remember the common formula to find HCF or LCM whenever "ratio" is given,

LCM= LCM of ratio x HCF.
2400(given) = 60 x HCF,
HCF = 40.

3x, 4x, 5x take LCM of these numbers is 60.
Formula:-numbers=LCM * hcf.
60x = 2400 * hcf,
X = 2400/60,
X = 40.

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.

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].

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.

Actually, the numbers given are in a ratio so we have to take in : 1/60.

So lcm:=1/2400,
and then,
The product of number=lcm of number*hcf of number;
1/60=1/2400*HCF.
HCF=40.

LCM = HCF*remaining factors.
or, 2400 = HCF *3 *4*5.
So, HCF = 2400÷60.
= 40.