Let's work through this step-by-step:
Step 1: We are given that there are 4 two-digit positive even integers and their sum is 86.
Step 2: Let's assume the largest number is y.
Step 3: Since we have four two-digit positive even integers, we can assume the other three numbers are a, b, and c, where a < b < c < y.
Step 4: We can write the equation for the sum of the four numbers as y + c + b + a = 86.
Step 5: Since y is the largest number, it should be as large as possible. Therefore, we need to maximize y.
Step 6: The largest two-digit even number is 98.
Step 7: Substituting the values into the equation: 98 + c + b + a = 86.
Step 8: Rearranging the equation: c + b + a = 86 - 98 = -12.
Step 9: Since the numbers are positive, we cannot have a negative sum. Therefore, the largest number cannot be 98.
Step 10: Let's try the next largest even number, which is 96.
Step 11: Substituting the values into the equation: 96 + c + b + a = 86.
Step 12: Rearranging the equation: c + b + a = 86 - 96 = -10.
Step 13: Again, we cannot have a negative sum, so 96 cannot be the largest number.
Step 14: Let's try the next largest even number, which is 94.
Step 15: Substituting the values into the equation: 94 + c + b + a = 86.
Step 16: Rearranging the equation: c + b + a = 86 - 94 = -8.
Step 17: Once again, we cannot have a negative sum, so 94 cannot be the largest number.
Step 18: Let's try the next largest even number, which is 92.
Step 19: Substituting the values into the equation: 92 + c + b + a = 86.
Step 20: Rearranging the equation: c + b + a = 86 - 92 = -6.
Step 21: Once again, we cannot have a negative sum, so 92 cannot be the largest number.
Step 22: Finally, let's try the next largest even number, which is 90.
Step 23: Substituting the values into the equation: 90 + c + b + a = 86.
Step 24: Rearranging the equation: c + b + a = \86 - 90 = -4.
Step 25: Again, we cannot have a negative sum, so 90 cannot be the largest number.
Step 26: Based on our calculations, we cannot find a positive even number that satisfies the equation. Therefore, it is not possible to determine the largest number from the given options.
Answer: None of the options (A) 30, (B) 40, (C) 50, (D) 58, (E) 64 can be the largest number.