# Reverse LCM/HCF

1. May 24, 2014

### tomtomtom1

Hi all

I was hoping someone could help answer the following question:-

I think of two numbers A & B

The HCF of A & B is 8
The LCM of A& B is 192

What are the numbers A&B?

I do the following:-

A*B = HCF & LCM
A*B = 8 * 192
A*B = 1536.

Prime factor (PF) both numbers to get:-

PF of 8 = 2 x 2 x 2
PF of 192 = 2 x 2 x 2 x 2 x 2 x 2 x 3

That is as far i can get, can someone help?

Thanks

2. May 24, 2014

### rasmhop

What makes you think there is a unique solution?

3. May 24, 2014

### tomtomtom1

i know that there could be several solutions but i am struggling to understand the process.

alos what could be the possbile numbers and how do you get to them?

4. May 24, 2014

### rasmhop

If you have no idea how to approach the problem I would suggest the following process:
Start listing the A's that could work in this problem, i.e. the A has 8 as a divisor and which itself is a divisor of 192. For instance your list might start:
A=8
A=16
...
For every such A ask yourself what B should be for their LCM and HCF to be what you wanted. This problem is small enough that you can do it for every A, but you will probably see the pattern before you are done.

EDIT: And remember to always look at prime decomposition. Never think of 16 or 32 think of 2^4 and 2^5.

EDIT2: The process I outlined is not an efficient process and there is an easy way to solve problems such as these if you know the right mathematical results about prime decomposition and have a little experience, but I think working it out manually is a good way to get a feel for the problem and I don't want to just give you the solution.

Last edited: May 24, 2014
5. May 24, 2014

### HallsofIvy

Staff Emeritus
I have no idea where you got this. It is certainly not true- the product of two numbers is NOT necessarily the product of the HCF and LCM. Saying that the "Highest common factor of A and B is 8 means that A= 8x and B= 8y for some integers x and y that have no further common factors. The "least common multiple" of A and B will be 8xy= 192 so that xy= 192/8= 24= (2^3)(3). Use the fact that x and y have no common factors to determine them.

Last edited: May 24, 2014
6. May 24, 2014

### lurflurf

from

HCF(A,B)=8
LCM(A,B)=192

next consider A/8 and B/8

what are

HCF(A/8,B/8)
LCM(A/8,B/8)

again problems of this type can have multiple solutions in general without further restrictions