Question

Find the number of positive integers n>2000 which can be expressed as n=2^m+2^n where m and n are integers (for example, 33=2^0+2^5)

Answer 1

Answer:

The number of positive integers are 65.

Step-by-step explanation:

To find : The number of positive integers n>2000 which can be expressed as $n=2^m+2^n$ where m and n are integers.

Solution :

As n>2000

We know that,

$2^{10}= 1024 < 2000 < 2048 = 2^{11}$

Since,

$2^{10}+2^{9} = 1536 < 2000$

Any combination of $2^m+2^n < 2000$

(as long as both m or n are not 10)

We would have the following possibilities,

$2^10 + 2^n , (n = 1,2….9)$

${2}^{9} + {2}^{n} , (n = 1,2….8)$

And so on;

The required number is given by,

$n=\frac{10\times 11}{2}$

$n = 65$

Therefore, the number of positive integers are 65.