What are the Numbers A and B Given HCF and LCM?

  • Context: High School 
  • Thread starter Thread starter tomtomtom1
  • Start date Start date
  • Tags Tags
    Reverse
Click For Summary

Discussion Overview

The discussion revolves around finding two numbers A and B given their highest common factor (HCF) of 8 and least common multiple (LCM) of 192. Participants explore the relationships between these numbers, including their prime factorization and potential methods for determining A and B.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant calculates that A * B = HCF * LCM = 1536 but is challenged on the validity of this approach.
  • Another participant questions the assumption of a unique solution for A and B, suggesting that multiple solutions may exist.
  • A suggestion is made to list possible values for A that are divisors of both 8 and 192, and to determine corresponding values for B based on the HCF and LCM conditions.
  • A later reply emphasizes the importance of prime decomposition and suggests considering A/8 and B/8 to analyze the problem further.
  • Participants note that without additional restrictions, problems of this type can have multiple solutions.

Areas of Agreement / Disagreement

Participants express differing views on the uniqueness of the solution, with some acknowledging the possibility of multiple solutions while others focus on the process of finding A and B. The discussion remains unresolved regarding the exact values of A and B.

Contextual Notes

Participants highlight the need for careful consideration of prime factors and the relationships between A, B, HCF, and LCM, indicating that assumptions about the uniqueness of solutions may not hold without further constraints.

tomtomtom1
Messages
160
Reaction score
8
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
 
Mathematics news on Phys.org
What makes you think there is a unique solution?
 
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?
 
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:
tomtomtom1 said:
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.
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.

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
 
Last edited by a moderator:
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
 

Similar threads

What Is the Other Number If HCF Is 33 and LCM Is 264?
  • · Replies 7 ·
Replies
7
Views
1K
MHB -c5.LCM and Prime Factorization of A,B,C
  • · Replies 3 ·
Replies
3
Views
1K
High School Nonsense can be truth in logic?
  • · Replies 84 ·
3
Replies
84
Views
8K
High School Fractional/decimal powers
  • · Replies 10 ·
Replies
10
Views
3K
High School Complete the square, yes, but what does it mean?
  • · Replies 19 ·
Replies
19
Views
4K
Undergrad What Is the Smallest Integer c for LCM(a, b, c) to Equal L?
  • · Replies 1 ·
Replies
1
Views
1K
High School Natural Numbers contain all the Primes
  • · Replies 24 ·
Replies
24
Views
3K
High School What is the process for finding the LCD of rational expressions and equations?
  • · Replies 21 ·
Replies
21
Views
3K
High School What went wrong with (-x)^2=x^2?
  • · Replies 22 ·
Replies
22
Views
3K
High School Popularizing a property for n-bonacci numbers without publishing it?
  • · Replies 12 ·
Replies
12
Views
2K