Find minimal value of N?

As N≡9 (mod 11) then N = 9+11x for some integer x

Take this and sub it into the middle line:
4 ≡ 9+11x (mod 9)
≡ 2x (mod 9)
So 2 ≡ x (mod 9)
so x = 2+9y for some integer y

Sub this back into N gives:
N = 9+11(2+9y) = 31 + 99y
Take this and sub it into the first line:
3 ≡ 31+99y (mod 7)
≡3 + y (mod 7)
So 0≡y (mod 7)

So minimum y is y = 0.
Sub this back into N give minimum N of 31.


If the question asked to find all N then the end of this question would be that y = 7z for some integer z and sub that in to give: N = 31 + 793z for some integer z.

31

Edit at other posts, since I doubt I'll get best with just the answer:
bskelkar: Your mods are entirely backwards. Check Ian's if you don't believe me.
Ian: Is that a typo? Should be N = 31 + 693z.

N-7 = 3p, N-9 =4q and N-11=9r, fro some integers p, q and r.
So 3p -3 =4q and 3p -4 = 9r.
I.e. 4q + 1 = 9r.So q = 2 and r = 1 or
q = 11, r = 5 or
q = 20, r = 9 or
q = 29, r = 13 ..... Go on trying till you get p as an integer.