476XY0 is divisible by both 3 & 11. The non-zero digits in the hundred's and ten's places are respectively.

To determine the non-zero digits in the hundred's and ten's places of the number 476XY0, we need to find the values of X and Y that make the number divisible by both 3 and 11.

Step 1: Divisibility by 3 A number is divisible by 3 if the sum of its digits is divisible by 3. In this case, the sum of the digits is 4 + 7 + 6 + X + Y + 0 = 17 + X + Y. For the number to be divisible by 3, the sum of the digits must be divisible by 3. Therefore, we have the equation 17 + X + Y ≡ 0 (mod 3).

Step 2: Divisibility by 11 A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is divisible by 11. In this case, the sum of the digits in odd positions is 4 + 6 + Y = 10 + Y, and the sum of the digits in even positions is 7 + X + 0 = 7 + X. For the number to be divisible by 11, the difference between these two sums must be divisible by 11. Therefore, we have the equation (10 + Y) - (7 + X) ≡ 0 (mod 11).

Step 3: Solving the equations To find the values of X and Y that satisfy both equations, we can try different values and check if they satisfy both equations simultaneously.

Let's start with X = 1 and Y = 2: 17 + 1 + 2 ≡ 0 (mod 3) => 20 ≡ 0 (mod 3) (not divisible by 3) (10 + 2) - (7 + 1) ≡ 0 (mod 11) => 4 ≡ 0 (mod 11) (not divisible by 11)

Let's try another set of values, X = 2 and Y = 1: 17 + 2 + 1 ≡ 0 (mod 3) => 20 ≡ 0 (mod 3) (not divisible by 3) (10 + 1) - (7 + 2) ≡ 0 (mod 11) => 2 ≡ 0 (mod 11) (not divisible by 11)

We can continue trying different values of X and Y until we find a combination that satisfies both equations.