Which positive integers Cannot be written as the sum of two or more consecutive numbers?
Table of Contents
- 1 Which positive integers Cannot be written as the sum of two or more consecutive numbers?
- 2 How do you prove a number is a power of 2?
- 3 How many ways can we write it as a sum of consecutive positive integers?
- 4 Why do powers of 2 not have consecutive sums?
- 5 How do you check if an integer is a power of 2?
Which positive integers Cannot be written as the sum of two or more consecutive numbers?
We can’t write every number as a sum of consecutive numbers – for example, 2, 4 and 8 can’t be written as sums of consecutive numbers. In the above, 9 and 15 were the only numbers that I could find that could be written in more than one way.
What numbers are not powers of 2?
Powers of 2 that contain no digits that are powers of 2
- Base 2 (binary): 1 and 0 are your digits and 1 is a power of 2, so 0 is the only digit you can use. This is a problem as the only number we can represent in binary with only the digit 0 is 0, which is not a power of 2.
- Base 3: 2,1, and 0.
- Base 4: 3,2,1 and 0.
How do you prove a number is a power of 2?
As the prime factorization of an integer is unique, n must be of the form 2r, where r≥0. Thus, n must be a power of 2. The sum of an arithmetic progression equals the number of terms times an integer or a half-integer. If twice the sum is a power of two, then both factors are a power of two.
How do you write two consecutive positive integers?
Solution: Let the two consecutive Numbers be x and x+1. Note : – 13 *14 = 182 , this is because I write 14x – 13x instead of x , so as to solve the quadratic equation . So the two consecutive positive Integers are 13 and 14 .
How many ways can we write it as a sum of consecutive positive integers?
Some numbers can be written as the sum of two or more consecutive positive integers, and some cannot. For example, 15 can be expressed in three different ways: 7+8, 4+5+6, and 1+2+3+4+5. But it is not possible to express 8 in this way….by David Radcliffe.
| 15 | −14 + … + 14 + 15 |
|---|---|
| 1 + 2 + 3 + 4 + 5 | 0 + 1 + 2 + 3 + 4 + 5 |
Which positive integers can be represented as the sum of two or more consecutive positive integers?
In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite. The impolite numbers are exactly the powers of two, and the polite numbers are the natural numbers that are not powers of two.
Why do powers of 2 not have consecutive sums?
Powers of 2 cannot be written as sums of consecutive whole numbers. You certainly cannot make them with an odd number r of consecutive whole numbers, since if you could r would be a factor of the number, and powers of 2 have no odd factors (apart from 1).
Are all powers of two even?
If it is known that n is indeed a power of 2, then only one bit in the integer can ever be lit. If the index of that bit is even, then your answer is even (assuming bit indices are 0-based). For negative numbers, you may want to use the absolute value.
How do you check if an integer is a power of 2?
Keep dividing the number by two, i.e, do n = n/2 iteratively until n becomes 1. In any iteration, if n\%2 becomes non-zero and n is not 1 then n is not a power of 2. If n becomes 1 then it is a power of 2.
How do you find consecutive positive integers?
If n is an integer, (n + 1) and (n + 2) will be the next two consecutive integers. For example, let n be 1. We find its consecutive integers as (1 + 1) and (1 + 2), or 2 and 3.