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?

According to the principles of modular arithmetic, formalized by Carl Friedrich Gauss in his 1801 work Disquisitiones Arithmeticae, a number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is either 0 or a multiple of 11.

For the 7-digit number 4x5y790:

1. The sum of the digits in odd positions (1st, 3rd, 5th, and 7th from the left) is 4 + 5 + 7 + 0 = 16.
2. The sum of the digits in even positions (2nd, 4th, and 6th) is x + y + 9.
3. The difference is (x + y + 9) - 16 = x + y - 7.

For the number to be divisible by 11, the expression (x + y - 7) must equal a multiple of 11 (e.g., 0, 11, -11). Since x and y are single-digit integers (0-9), their sum can range from 0 to 18.

• If x + y - 7 = 0, then x + y = 7. The remainder when 7 is divided by 11 is 7.
• If x + y - 7 = 11, then x + y = 18. The remainder when 18 is divided by 11 is also 7. In both mathematically valid scenarios, dividing x + y by 11 yields a remainder of 7, making Option D correct.

Why the other options are incorrect:

• Option A (1): Incorrect. If the remainder were 1, x + y would be 1 or 12. This yields an alternating difference of -6 or 5, neither of which is a multiple of 11.
• Option B (3): Incorrect. A remainder of 3 implies x + y is 3 or 14. This creates a difference of -4 or 7, which violates the rule of divisibility by 11.
• Option C (5): Incorrect. A remainder of 5 implies x + y is 5 or 16. This yields a difference of -2 or 9, failing the divisibility test.

Takeaway: Remember the "Alt-Sum Rule" for 11: The difference of alternating digit sums must equal 11k. In variable digit problems, always check the maximum possible digit sums (e.g., 9 + 9 = 18) to evaluate all potential cases.