for the 1st one:

A number when divided by 7 leaves remainder r  (0,1,2,3,4,5,6).

So for a triad of numbers, the traid of remainders has 7*7*7 = 343 possibilties.

 

Now, the squares of numbers leave remainders 0,1,2 or 4 as remainders.

 

Hence if a²+b²+c² is div by 7 then either a,b and c are div by 7 or the remainders are permuations of (1,2,4).

 

The first case has one member in the set of triads of remainders.

 

Now, lets consider a²  1; b²  2 and c²  4 (mod 7)

 

You can check that number of favourable remainder triads is 8 for this case. and there are 6 such permutations of (1,2,4). Hence our count of favourable triads is 1+6*8 = 49.

 

Hence, the required prob = 49/343 = 1/7.