Find X and Y - Show working out please

Question

x and y are two numbers each greater than 3,

 

the highest common factor of x and y is 3,

the lowest common multiple of x and y is 36,

find x and y

Answer 1

 

We need 2 numbers that meet this critria:

  (1) Both numbers are larger than 3

  (2) Both numbers are multiples of 3

  (3) The LCM of both numbers is 36 

  (4) 3 is the largest common factor of both numbers

 

So the numbers have to be between 3 and 36

 

The numbers that meet the above criteria are:

 

   9  and  12

 

CHECK:

   Factors of 9   ... 1, 2, 3, 9

   Factors of 12 ... 1, 2, 3, 4, 6, 12

 

   The largest common factor is 3

   

   Multiples of   9 ... 9, 18, 27, 36, ...

   Multiples of 12 ... 12, 24, 36, ...

 

   The least common multiple is 36

 

Answer 2

x and y are two numbers each greater than 3,
the highest common factor of x and y is 3,
the lowest common multiple of x and y is 36,
find x and y

xy = product of greatest common divisor and least common multiple=3×36=108

Since GCD is 3 we may write

x=3a, y=3b

since both x, y are greater than 3, we have a, b are both greater than 1

xy=9ab and so ab=12  The only factors that are both greater than 1 are 2 and 6 or 3 and 4

for 2 and 6 we have x=6 y=18, but their GCD is 6 not 3.

for the other pair we have x=9, y=12, GCD =3.  We are done.