Question 1052975
Let x be this unknown number that satisfies all three conditions given. For the sake of simplicity, we'll assume x is positive. 


Use the <a href="http://www-math.ucdenver.edu/~wcherowi/courses/m5410/crt.pdf">Chinese Remainder Theorem</a>


I will use the same basic steps and notation as that PDF is showing. 


We have these 3 congruences given
x = 1 (mod 7)
x = 2 (mod 9)
x = 3 (mod 11)


The {{{a[i]}}} values are the right hand side values (1, 2 and 3 respectively)  where 'i' refers to the modulus value
{{{a[7] = 1}}}
{{{a[9] = 2}}}
{{{a[11] = 3}}}



Multiply out the modulus values
N = 7*9*11 = 693


Calculate the {{{M[i]}}} values
{{{M[7] = N/7 = 693/7 = 99}}}
{{{M[9] = N/9 = 693/9 = 77}}}
{{{M[11] = N/11 = 693/11 = 63}}}


Then finally we need to compute the multiplicative inverses {{{y[i]}}} values of each {{{M[i]}}} value


{{{y[7] = (M[7])^(-1) = (99)^(-1) = 1}}} (mod 7)
Note: 99 = 1 (mod 7) so the solution to 1y = 1 (mod 7) is y = 1


{{{y[9] = (M[9])^(-1) = (77)^(-1) = 2}}} (mod 9)
Note: 77 = 5 (mod 9). The solution to 5y = 1 (mod 9) is y = 2


{{{y[11] = (M[11])^(-1) = (63)^(-1) = 7}}} (mod 11)
Note: 63 = 8 (mod 11) and the solution to 8y = 1 (mod 11) is y = 7



Now we use the information computed above to get the solution for x


{{{x = a[7]*M[7]*y[7] + a[9]*M[9]*y[9] + a[11]*M[11]*y[11]}}} (mod 693)


{{{x = 1*99*1 + 2*77*2 + 3*63*7}}} (mod 693)


{{{x = 1730}}} (mod 693)


{{{x = 344}}} (mod 693)


The smallest positive solution for x is <font color=red>344</font>


The set of all solutions for x is the set of x values such that {{{x = 693t+344}}} where {{{t>=0}}} and {{{t}}} is an integer. In other words, <font color=red>x = 344 (mod 693)</font>