Number Systems

We are asked to find the least negative integral value of such that the expression is divisible by 64 for all .
To solve this, let's analyze the powers of 49 and 16 modulo 64.
Step 1: Simplify modulo 64
Note that: So, we calculate the powers of 49 modulo 64: It is clear that will repeat periodically modulo 64, and we need to find a pattern for any general value of .
Step 2: Simplify modulo 64
We now calculate powers of 16 modulo 64: For higher powers of 16, will be congruent to 0 modulo 64 for .
Step 3: Analyze the expression
We now combine and modulo 64. For , , so the expression simplifies to: For the expression to be divisible by 64, we need: Using the fact that , , and so on, we can see that the least value of that will make this true is .
Thus, the least negative integral value of is .