Let, the numbers be "x" and "y". Consider "x" as the triple digit number.
Now, according to the question,
x + y = 217..........................equation (1)
x * y = 642............................equation (2)
Now,
Taking equation (2),
x * y = 642
y = 642 / x..................................equation (3)
Now, Taking equation (1),
[tex]x+y=217[/tex]
Substituting the value of y from equation (3), we get,
[tex]x+ \frac{642}{x} =217[/tex]
[tex]\frac{x *x+642}{x} =217[/tex]
[tex] x^{2} +642 =217*x[/tex]
[tex] x^{2} +642 =217x[/tex]
[tex] x^{2} +642-217x =0[/tex]
[tex] x^{2} -217x+642 =0[/tex]
[tex] x^{2} -3x-214x+642 =0[/tex]
[tex]x(x-3)-214(x-3)=0[/tex]
[tex](x-3)(x-214)=0[/tex]
Using zero product property,
EITHER,
x - 3 = 0
x = 3
OR,
x - 214 = 0
x = 214
Since, "x" is the triple digit number, x = 214.
Now,
Taking equation (2),
x * y = 642
Substituting the value of "x" in the equation, we get,
(214) * y = 642
y = 642 / 214
y = 3
So, the numbers are 214 and 3.
