find two natural numbers a and b given their greatest common divisor D=12 and their least common multiple M=432 give - Maths - Number Systems

Question

find two natural numbers a and b given their greatest common
divisor D=12 and their least common multiple M=432 give the method
of finding solutions in the general case

Ajanta Trivedi · Accepted Answer

HCF of two numbers = $12 = 2^{2} * 3$ and

LCM of two numbers = $432 = 2^{4} * 3^{3}$

let the number be a and b . and the numbers must be of the form $2^{m} * 3^{n}$

where $2 \leq m \leq 4 a n d 1 \leq n \leq 3$ [since HCF is the common factor]

if a = $2^{2} * 3 = 12$ $\Rightarrow b = 2^{4} * 3^{3} = 432$

if $a = 2^{4} * 3 = 48 \Rightarrow b = 2^{2} * 3^{3} = 108$

if $a = 2^{2} * 3^{3} = 108 \Rightarrow b = 2^{4} * 3 = 48$

if $a = 2^{4} * 3^{3} = 432 \Rightarrow b = 2^{2} * 3 = 12$

hope this helps you.

cheers!!

find two natural numbers a and b given their greatest common divisor D=12 and their least common multiple M=432.give the method of finding solutions in the general case.