Math Problem Statement
Let n >= 1 be an integer. Prove that n can be expressed in the form : n = Σc_k * k! , from k = 1 to m where each c_k is an integer and 0 <= c_k <= k. For example, 44 = 1*(4!) + 3*(3!) + 1*(2!) + 0*(1!). Hint: Use induction and apply the division algorithm to divide n+1 by m! for a suitable m.
Solution
To prove that every integer can be expressed in the form where each is an integer and , we'll use mathematical induction and the division algorithm.
Base Case
For : We need to express in the form where for each .
Since , we can take . Thus, the base case holds with and .
Inductive Hypothesis
Assume that for some integer , we can express in the form where for all .
Inductive Step
We need to show that can also be expressed in the same form.
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Choose : Let be the largest integer such that . According to this choice, we have:
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Apply the Division Algorithm: Divide by using the division algorithm. This gives: where and is an integer.
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Check Bounds for : Since , it follows that .
Thus, we have .
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Construct the Representation for : Write Since , by the inductive hypothesis, can be written as a sum of factorial terms in the form where for each .
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Combine Terms: Substitute the expression for back into the equation for : This expression represents as a sum of factorial terms, with coefficients for and , satisfying for all .
Therefore, by induction, every integer can be expressed in the form where for each .
Conclusion
This completes the proof by induction. The result follows that any integer can be expressed as a sum of factorials with the given constraints on the coefficients .
Would you like further details on any part of this proof or have additional questions?
- How does the division algorithm help in breaking down numbers?
- Why do we choose such that ?
- What is the significance of having ?
- How does this approach ensure uniqueness in the representation?
- Can this factorial representation be applied to negative numbers?
Tip: Inductive proofs often simplify complex problems by establishing a base and building incrementally from it. This can help in breaking down abstract concepts into manageable parts.
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Math Problem Analysis
Mathematical Concepts
Mathematical Induction
Factorials
Division Algorithm
Formulas
n = Σc_k * k!
Theorems
Inductive Step
Division Algorithm
Suitable Grade Level
Grades 10-12