Math Problem Statement
Solution
Let's go through these questions one by one:
1. Montrer que la somme de trois entiers consécutifs est divisible par 3.
We can represent the three consecutive integers as , , and . The sum of these integers is:
Since is divisible by 3, the sum of three consecutive integers is always divisible by 3.
2. Montrer de même que la somme de cinq entiers consécutifs est divisible par 5 et que la somme de sept entiers consécutifs est divisible par 7.
For five consecutive integers:
Represent the integers as , , , , and . The sum is:
Thus, the sum is divisible by 5.
For seven consecutive integers:
Represent the integers as , , , , , , and . The sum is:
Thus, the sum is divisible by 7.
3. Montrer que la somme de six entiers consécutifs n'est jamais divisible par 6.
Represent the six consecutive integers as , , , , , and . The sum is:
Since , the sum will not be divisible by 6 because of the remainder 15, which leaves 3 modulo 6.
4. Plus généralement, on étudie la somme de entiers consécutifs en notant le plus petit d'entre eux.
Let the integers be , , , ..., .
4a. Réduire cette somme à l'aide des suites arithmétiques.
The sum of the first consecutive integers starting at is:
The sum of the first integers is , so:
4b. Montrer que cette somme est divisible par si et seulement si est impair.
The sum is:
For this to be divisible by , the second term must also be divisible by . This happens if is odd, as is even, making an integer, which ensures the entire expression is divisible by .
Would you like a deeper explanation for any part of this solution?
Here are five related questions:
- How would the formula change for -consecutive numbers starting from any non-integer?
- What if the sequence skips values, e.g., taking only even integers?
- How would we handle divisibility for a sequence of powers (e.g., squares or cubes)?
- Is there a general rule for when sums of any sequence of integers are divisible by the count?
- How can these sums be related to the properties of modular arithmetic?
Tip: When working with consecutive integers, grouping terms can help reveal patterns of divisibility.
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Math Problem Analysis
Mathematical Concepts
Divisibility
Consecutive Integers
Arithmetic Sequences
Formulas
Sum of n consecutive integers: S_k = k·n + (sum of first k-1 integers)
Sum of first (k-1) integers: (k-1)k/2
Theorems
Divisibility Rule: A sum is divisible by k if k divides both the sum of the integers and the sequence properties allow divisibility.
Suitable Grade Level
Grades 10-12