If x and y are the two digits of the number 653xy such that this number is divisible by 80, then what is the value of x + y?

Introduction / Context:
This problem involves divisibility by 80 for a five digit number with two unknown digits. Divisibility by 80 can be analyzed using its prime factorization and standard divisibility rules for powers of 2 and for 5. The task is to find digits x and y in the number 653xy so that it becomes divisible by 80 and then compute x + y. This tests understanding of divisibility rules and handling of unknown digits in place value notation.

Given Data / Assumptions:

- The number has the form 653xy, where x is the tens digit and y is the units digit. - The number is divisible by 80. - x and y are digits from 0 to 9. - We must find x + y.

Concept / Approach:
First factorize 80. We have 80 = 2^4 * 5. A number divisible by 80 must be divisible by 16 and by 5 simultaneously. Divisibility by 5 requires the last digit to be 0 or 5, but because 80 is also a multiple of 10, any multiple of 80 must end in 0. Therefore, y must be 0. Then, for divisibility by 16, the last four digits of the number must form a number divisible by 16. Hence we examine the four digit number 53x0 and find which value of x makes it divisible by 16.

Step-by-Step Solution:
Step 1: Write the number as 653xy. For divisibility by 5 and 10, the last digit y must be 0 for a multiple of 80, so set y = 0. Step 2: The number is now 653x0. The last four digits are 53x0, which must be divisible by 16. Step 3: Consider possible values of x from 0 to 9 and test divisibility of 53x0 by 16. Step 4: Instead of testing blindly, note that 5300, 5310, 5320 and so on will vary in steps of 10. We seek 53x0 such that 53x0 ÷ 16 is an integer. Step 5: One reliable method is to test directly. For x = 6, the number is 5360. However our full number is 65360, whose last four digits are 5360, and 5360 ÷ 16 = 335 exactly. Step 6: Checking for other x values shows that none of them give a last four digit block divisible by 16. Therefore x = 6 is the unique solution. Step 7: We already have y = 0 from the divisibility by 80 condition. Step 8: Compute x + y = 6 + 0 = 6.

Verification / Alternative check:
Verify that 65360 is divisible by 80. Compute 65360 ÷ 80. This is the same as 65360 ÷ (16 * 5). First divide by 16 to get 4085, then divide 4085 by 5 to get 817. Since both divisions are exact, 65360 is indeed divisible by 80. Any change to x or y would break at least one of the divisibility conditions, confirming that x = 6 and y = 0 is the only suitable pair.

Why Other Options Are Wrong:
If x + y were 5, 4 or 3, then the sum of the digits would correspond to a different pair such as (5,0), (4,0) or (3,0) or similar combinations. None of these yield a number 653xy that is divisible by both 16 and 5 in the required way. Thus only the sum 6 is consistent with all the constraints.

Common Pitfalls:
Some students may incorrectly use divisibility by 8 and 10 instead of by 16 and 5, which guarantees divisibility by 40 rather than 80. Others may assume that a last digit of 5 is acceptable, forgetting that multiples of 80 must end with 0. Paying attention to prime factorization and correct divisibility rules avoids these mistakes.

Final Answer:
The value of x + y is 6, which corresponds to option A.