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

Concept: For expressions of the form \[ (a+b\sqrt m)^n \] we use the conjugate \[ (a-b\sqrt m)^n. \] The sum of these two expressions is always an integer.

Step 1: Analyse \(x=(15+7\sqrt5)^9\).
Consider \[ (15+7\sqrt5)^9+(15-7\sqrt5)^9. \] This is an integer. Also, \[ |15-7\sqrt5|<1. \] Therefore, \[ 0<(15-7\sqrt5)^9<1. \] Hence \[ x = N-\varepsilon, \] where \(N\) is an integer and \(0<\varepsilon<1\). Thus, \[ [x]=N-1. \] The integer \(N\) is odd, therefore \[ [x] \] is even.

Step 2: Analyse \(y=(13+5\sqrt7)^{11}\).
Similarly, \[ (13+5\sqrt7)^{11} + (13-5\sqrt7)^{11} \] is an integer. Also, \[ |13-5\sqrt7|<1. \] Hence, \[ [y]=M-1, \] where \(M\) is an even integer. Therefore, \[ [y] \] is odd. Hence, \[ [x] \text{ is even and } [y] \text{ is odd}. \] \[ \boxed{\text{Even integer and odd integer respectively}} \]