Question

Let \([t]\) represent the greatest integer less than or equal to \(t\). If \[ x = (7\sqrt{5} + 15)^9 \quad \text{and} \quad y = (5\sqrt{7} + 13)^{11}, \] then \([x]\) and \([y]\) are:

• A. Even integer and odd integer respectively
• B. Odd integer and even integer respectively
• C. Both odd integers
• D. Both even integers

Hint:

If the conjugate term has magnitude less than 1, then the original surd expression is just less than a nearby integer. This trick is extremely useful in greatest integer function problems.

Answer 1

The Correct Option is A