If x and y are two digits and the number 4x5y790 is divisible by 11, then what is the remainder, if x + y is divided by 11?

Correct Option: (b) 3

The given number is 4x5y790 and it is divisible by 11. The divisibility rule for 11 states that the difference between the sum of digits at odd places and the sum of digits at even places must be divisible by 11 (i.e., 0, 11, 22, etc.). Let's number the digits from right to left starting from 1: 0(1st), 9(2nd), 7(3rd), y(4th), 5(5th), x(6th), 4(7th). Sum of digits at odd places (1st, 3rd, 5th, 7th) = 0 + 7 + 5 + 4 = 16. Sum of digits at even places (2nd, 4th, 6th) = 9 + y + x = 9 + x + y. The difference between these two sums is (16) - (9 + x + y) = 7 - (x + y). For this to be divisible by 11, 7 - (x + y) must be a multiple of 11. Since x and y are single digits (0 to 9), x + y ranges from 0 to 18. The only possible value for 7 - (x + y) to be a multiple of 11 is -4 (which is not possible for digits) or if 7 - (x + y) = -11, then x + y = 18. Wait, let's re-verify: Difference = (16) - (9 + x + y) = 7 - x - y. If x + y = 7, difference is 0, which is divisible by 11! If x + y = 7, then when x + y is divided by 11, the remainder is 7? Wait, let's check: 7 divided by 11 gives quotient 0 and remainder 7. But wait, let's re-evaluate options: (a) 1, (b) 3, (c) 5, (d) 7. If x+y = 7, remainder is 7. Let's check if another sum is possible. If x + y = 18, 7 - 18 = -11, which is divisible by 11. If x + y = 18, 18 divided by 11 gives remainder 7. Wait, is there any other combination? Let's check option (a) 1. If remainder is 1, x+y could be 12 (12 - 7 = 5, not mult of 11). Let's check 7 - (x+y) = -4, impossible. Wait, let's re-add: odd places from right: 0 (1st), 7 (3rd), 5 (5th), 4 (7th). Sum = 16. Even places: 9 (2nd), y (4th), x (6th). Sum = 9 + x + y. Difference = 16 - (9 + x + y) = 7 - x - y. For 7 - x - y to be a multiple of 11, 7 - x - y = 0 => x + y = 7. Or 7 - x - y = -11 => x + y = 18. In both cases, x + y = 7 or 18. If x + y = 7, remainder when divided by 11 is 7. If x + y = 18, 18 / 11 leaves remainder 7. Thus, the remainder is 7.