Question

the sum of two numbers is 384.What are the possible number if their is HCF 48?

Accepted Answer

Take note that 384 divided by 48 = 8. 
Hence, if we let one of the numbers as 48, the other number which is 
384 - 48 = 336 will still be divisible by 48. 
Hence, the two numbers are 48 and 336.

Answer

Let number be x and y ( positive integer).
So by given condition 
x+y=384     (i)
since HCF of x and y  is 48
therefore
x xx y=m(48)^2     (ii)   m is an integer.
x={m(48)^2}/y           (iii)
substitute x from (iii) in (i)
{m(48)^2}/y+y=384
m(48)^2+y^2=384y     (iv)
let  y=48l
therefore    (iv) reduces to
m(48)^2+(48l)^2-384xx48l=0
l^2-8l+m=0         (v)
solving (v) for l and m ,we may get many solutions.
m=16 ,l= 4 then  y=192, x=192
m=15 ,l=3 then  y=144 x=240
m=12 ,l= 2 then  y= 96 ,x= 288
m=7 ,l=1 then    y=48, x=336

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the sum of two numbers is 384.What are the possible number if their is HCF 48?

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