# Problem of the Day

Let and be positive integers such that . If Â , then could equal all of the following except:

A. 33

B. 65

C. 123

D. 257

E. 513

### Solution

[latexpage]

$k\cdot2^n=2^y+2^n$

If $y>n$, then we can factor out a $2^n$ from the right side:

$k\cdot2^n=2^n(2^{y-n}+1)$

$k=2^{y-n}+1$

Notice that this says that $k$ is equal to one more than a power of 2. Every answer choices is one more than a power of two except 123.

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