Question

if 814y is divisible by 6 where y is a digit find values of y

Answer 1

Group the number NN into blocks of three, and alternately assign +1+1 and −1−1 to these blocks starting with +1+1 to the right most block. Now multiply each of the blocks by the assigned numbers and add. If this sum is SS, then N≡S(mod11)N≡S(mod11). In other words, 11∣N11∣N if and only if 11∣S11∣S.

The more well known method is the one in which you do the same procedure on blocks of length one, with the same conclusion, including the remainder.

To illustrate with the given N=8276⋆845N=8276⋆845,

N≡S=845−76⋆+82=167−⋆≡2−⋆(mod11)N≡S=845−76⋆+82=167−⋆≡2−⋆(mod11).

Thus 11∣N11∣N if and only if ⋆=2⋆=2.

In view of my opening comments, the same choice of ⋆⋆ also makes NNdivisible by both 77 and 1313. 

Answer 2

There are two conditions necessary for the divisibility by 6
1) It must be divided by 2
2) It must be divided by 3
so the last digit must be even and the sum of all the numbers is equal to a no. which is correctly divided by 3
let us try 8+1+4+ y = 13 +y
=13 +2 =15
=13 +8 =21
the last digit must be 2 or 8 .