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.

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.

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.