A pair of numbers has a GCF of 14 and LCM of 168. What could the numbers be?

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  • Amy
    Lv 7
    1 month ago

    GCF is 14, so the numbers are 14*A and 14*B, where A and B have no factors in common.

    LCM of 14*A and 14*B is 14*A*B.

    A * B = 12

    A and B could be 1 and 12 or they could be 3 and 4. They cannot be 2 and 6 which have a factor in common.

    Thus your two numbers are 14 and 168 or 42 and 56.

  • 1 month ago

    56,84

    has GCF 14 and also LCM 168

  • 1 month ago

    Greatest Common Factor is 14, so both numbers must have a factor of 14 (or 7 and 2, as prime factors).

    Least Common Multiple is 168 = 14 * 12 = 14 * 3 * 2 * 2

    which means that the factors 3, 2, and 2 must be distributed among the numbers, and 2 cannot be in both numbers, as it would increase the GCF value.  The ways of distributing the factors are

    3,2,2 and (1); or

    3 and 2,2

    The first means that 12 is applied as a factor to one number, yielding 12 x 14 = 168, the other number being 1 x 14 = 14.

    The second case has the first number as 3 x 14 = 42, and the other as 2 x 2 x 14 = 56.

    So the complete answer is either 14 and 168, or 42 and 56.

    -----

    Fast way for competitions:

    If they only need one possible pair, then the numbers ARE the GCF and LCM.  That will work for any GCF and LCM.

  • Elaine
    Lv 7
    1 month ago

    Here is one method  I tend to do inspection as I recognise the patterns in  the multiplication tables 

    12 X 12 = 144

    168 - 144 = 24 

    168 = 12 X 14

     14 = 14 X 1

    The numbers are 14 and 12. 

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  • 1 month ago

    Let x,y be the pair of numbers, then

    xy=14(a)*14(b)=14*168, where (a,b)=1

    =>

    (14^2)ab=14*168

    =>

    ab=168/14=12

    =>

    a | 1 | 3 | 4 |12|

    b |12| 4 | 3 | 1 |

    =>

    x |14...|42...|56...|168 |

    y |168 |56...|42...|14...|

    =>

    the pair of numbers are

    14 & 168

    or

    42 & 56

  • 1 month ago

    168 = 14 x 12   

  • 1 month ago

    A pair of numbers has a GCF of 14 and LCM of 168.

    12 = 2^2 × 3

    14 = 2 × 7

    42 and 84 are the two numbers.

  • 1 month ago

    GCF of 14. so (1) = a*14 and (2) = b*14

    lcm of 168 so 168 = 14*12 = 2^3 *3 * 7

    14 = 2*7

    12 = 3*4

    so one needs to be 2*7 *3 = 42

    and the other must be 2*2*7 = 28

  • 168 = 2^3 * 3 * 7

    1 , 2 , 3 , 4 , 6 , 7 , 8 , 12 , 14 , 21 , 24 , 28 , 42 , 56 , 84 , 168

    The numbers must be greater than 14 and one can't divide evenly into another, and they must be divisible by 14 as well.  In order to avoid being divisible by 28, one of them must be divisible only by 2 and not by 4

    28 , 42 , 56 , 84 , 168

    42 and 28

    42 and 56

    That's it.

    42 and 84 can't work because 84/42 = 2, so they have a GCF of 42

    42 and 168 can't work for the same reason

  • alex
    Lv 7
    1 month ago

    Hint:

    168 = 14 x 12   

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