Question 3
For 
n
∈
N
0
n∈N 
0
​
  let 
w
(
n
)
w(n) denote the number of 
1
1s in the binary representation of 
n
n. For example, 
w
(
9
)
=
2
w(9)=2, since $9$ is 
1001
1001 in binary. Try to find a closed formula for 
g
(
n
)
g(n) in terms of 
n
n and 
w
(
n
)
w(n). If you succeed, the following question will be very easy.

Let 
n
=
10000000000000011
n=10000000000000011 in binary notation. What is 
g
(
n
)
g(n)? Write your answer in binary!

0 / 1 point
Incorrect
4.
Question 4
Let 
n
=
1152921504606846987
n=1152921504606846987. If we compute 
(
n
k
)
m
o
d
 
 
2
( 
k
n
​
 )mod2 for 
k
=
0
…
7
k=0…7, which sequence do we get? 

Write your answer as a simple 0/1 string. For example, if your sequence is 
(
1
,
0
,
1
,
0
,
1
,
0
,
1
,
0
)
(1,0,1,0,1,0,1,0), answer 
10101010
10101010.

Let's break down and solve the questions one by one.

### Question 3: Finding $ g(n) $

Given:
- $ n \in \mathbb{N}_0 $
- $ w(n) $ denotes the number of 1s in the binary representation of $ n $.

We need to find a closed formula for $ g(n) $ in terms of $ n $ and $ w(n) $.

Let's analyze the problem by first understanding $ g(n) $. In many problems involving binary representations, a common pattern is counting specific bit configurations or analyzing parity.

For example, consider the function $ g(n) = n \times 2^{w(n)} $. This function would combine $ n $ with the number of 1s in its binary form. Let's now try to apply this and see what happens when we use a specific $ n $.

Given $ n = 10000000000000011_2 $ (in binary):

1. Convert $ n $ to decimal:
   \[
   n = 2^{16} + 2^1 + 2^0 = 65536 + 2 + 1 = 65539
   \]
   
2. Calculate $ w(n) $, the number of 1s in the binary representation:
   \[
   w(65539) = 3
   \]
   
3. Let's assume $ g(n) = n \times 2^{w(n)} $ as a possible formula. Substituting the values:
   \[
   g(65539) = 65539 \times 2^3 = 65539 \times 8 = 524312
   \]
   
4. Convert $ 524312 $ back to binary:
   \[
   g(65539) = 1000000000010101000_2
   \]

This would be the result, but it does not necessarily match any specific answer format expected. Thus, let's revisit the definition of $ g(n) $.

### Understanding $ g(n) $
Since we are asked to relate $ g(n) $ with $ w(n) $, consider another simpler function like $ g(n) = n \times w(n) $. This will likely involve a modular reduction or additional operation. Let’s try to simplify:

### Question 4: Sequence from Binomial Coefficients Modulo 2

Given $ n = 1152921504606846987 $:

We need to calculate $ \binom{n}{k} \mod 2 $ for $ k = 0, 1, 2, \ldots, 7 $.

#### Steps:
1. Convert $ n $ to binary:
   \[
   n = 1152921504606846987 = 1000000000000000000000000000000011_2
   \]
   
2. For each $ k $, determine $ \binom{n}{k} \mod 2 $:

   Use Lucas' theorem, which states:
   \[
   \binom{n}{k} \mod 2 = \prod \binom{n_i}{k_i} \mod 2
   \]
   where $ n_i $ and $ k_i $ are the binary digits of $ n $ and $ k $, respectively.
   
3. The sequence $ \binom{n}{k} \mod 2 $ for $ k = 0 $ to $ 7 $ corresponds to the binary digits of $ n $ restricted to the first 3 digits (since the binomial coefficient of any bit higher than $ k $ would cancel out):
   
   - For $ k = 0 $, $ \binom{n}{0} = 1 $.
   - For $ k = 1 $, $ \binom{n}{1} = 1 $ (if the least significant bit is 1).
   - Continue for all $ k = 0 $ to $ 7 $.

The sequence is therefore:
\[
\text{Sequence} = 11000000
\]

This sequence comes from the fact that $ n $ has 1s in the first and second least significant positions.

### Summary:
- **Question 3**: The binary result was an attempt based on a function $ g(n) $ with the hypothesis that could be incorrect as it does not yield standard expected results.
- **Question 4**: The correct answer sequence is `11000000`.

Would you like to go through the steps in more detail or explore further examples?

---

**Relative Questions**:
1. How can Lucas' theorem be generalized for other moduli?
2. What are the properties of binary representations when considering bit manipulation?
3. What would $ g(n) $ be if we modified the problem to use different weights?
4. What happens if $ w(n) $ is a very large number, like in the case of very dense binary numbers?
5. How does one efficiently compute large binomial coefficients modulo small primes?

**Tip**: Always try decomposing complex binary or modular problems using smaller, simpler examples to verify your hypotheses.

Question 3: For n ∈ N0, let w(n) denote the number of 1s in the binary representation of n. For example, w(9) = 2, since 9 is 1001 in binary. Try to find a closed formula for g(n) in terms of n and w(n). If you succeed, the following question will be very easy. Let n = 10000000000000011 in binary notation. What is g(n)? Write your answer in binary. Question 4: Let n = 1152921504606846987. If we compute (n choose k) mod 2 for k = 0…7, which sequence do we get? Write your answer as a simple 0/1 string.

Explore problems involving binary representation and modulo 2 binomial coefficients. Learn how to determine the number of 1s in a binary number and apply Lucas' theorem to compute sequences mod 2. Suitable for students in Grades 11-12.

Math Problem Statement

Question 3 For n ∈ N 0 n∈N 0 let w ( n ) w(n) denote the number of 1 1s in the binary representation of n n. For example, w ( 9 )

Let n

0 / 1 point Incorrect 4. Question 4 Let n

1152921504606846987 n=1152921504606846987. If we compute ( n k ) m o d 2 ( k n )mod2 for k

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation