Question

In this question x^y stands for x raised to the power y .for example 2^3=8 and 4^1.5=8

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: 65​

Answer 1

Answer:

65

Step-by-step explanation:

In this question x^y stands for x raised to the power y .for example 2^3=8 and 4^1.5=8 Find the number of positive integers n < 2000 which can be expressed as n=2^m+2^n where m and n are integers

Number of positive integers < 2000

2^m & 2ⁿ can be

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32

2⁶ = 64

2⁷ = 128

2⁸ = 256

2⁹ = 512

2¹⁰ = 1024

2¹¹ > 2000

with 2⁰ = 1  all 11 numbers can be added

with 2¹ = 2   10 numbers can be added except 2⁰ = 1 which is already considered

Similarly

till 2⁹ = 512   , 2 numbers can be added

for 2¹⁰ = 1024 , no additional numbers can be added as it will make number > 2000

11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2

Sum = (10/2)(11 + 2)

= 5 * 13

= 65