Question

Prove the following statements using any method from Chapters 4,5 or 6 . The product of any five consecutive integers is divisible by 120 . (For example, the product of 3,4,5,6 and 7 is 2520 , and \(2520=120 \cdot 21\).

Short answer

The product of any five consecutive integers is divisible by 120 because within any sequence of five consecutive integers, there must be at least one integer divisible by 2, one integer divisible by 3, and one integer divisible by 5. Since 120 is the product of 2, 3, and 5, the product of any five consecutive integers will be divisible by 120.

Step-by-step solution

Represent Five Consecutive Integers

Let us represent any five consecutive integers as \(n\), \(n+1\), \(n+2\), \(n+3\), \(n+4\). Here, \(n\) can be any integer.

Calculate the Product

The product of these five consecutive integers can be written as \(n(n+1)(n+2)(n+3)(n+4)\).

Analyze the Factors

Now, let's analyze the factors. In these five consecutive integers, there must be at least one number divisible by 2, at least one number divisible by 3, and at least one number divisible by 5. This is because the sequence of five consecutive numbers necessarily includes numbers divisible by these values.

Conclusion

Since the product of the numbers is divisible by 2, 3, and 5, it is necessarily divisible by the product of these numbers, which is 120. Thus, we have proved that the product of any five consecutive integers is divisible by 120.

Key concepts

Consecutive Integers

Consecutive integers follow one after the other in sequence without any gaps. For example, 3, 4, 5, 6, and 7 are consecutive integers because there's a consistent increment of one between each number.

Understanding consecutive integers is crucial when tackling problems that require you to consider the properties of a series of numbers, such as their divisibility. When looking at a product of consecutive integers, we're essentially multiplying a sequence of numbers that are spaced evenly on the number line. This forms the basis for analyzing divisibility, as certain patterns emerge that can predictably indicate whether the product will be divisible by a particular number, such as 120 in our exercise.

Proof Techniques

In mathematics, proof techniques refer to the various methods used to demonstrate the truth of a statement. Direct proof, proof by contradiction, and mathematical induction are among the common methods.

For the problem at hand, we are considering a direct proof where we show that a property - divisibility by 120 - holds true by leveraging the inherent features of consecutive integers. We essentially dissect the problem piece by piece to establish that, no matter which integer we start with, the resulting product will always have the required factors to be divisible by 120.

Factor Analysis

Factor analysis in mathematics involves breaking down numbers into their prime constituents. When examining the product of five consecutive integers, factor analysis helps us to identify the prime numbers that will invariably be present in the sequence.

Considering that any five consecutive numbers will span at least one multiple of 2, 3, and 5, these primes become the critical factors. Since 120 is equal to \(2^3\cdot3\cdot5\), we need to establish that these primes are represented within our product. By analyzing the five numbers, we show that the requisite multiple of 2 (at least two of them), 3, and 5 are inherently included due to the nature of consecutive integers, ensuring the product is divisible by 120.

Mathematical Induction

Mathematical induction is a powerful proof technique used to establish the validity of a statement for all natural numbers. It proceeds in two steps: the base case, which tests the statement for the initial number, and the inductive step, where we assume the statement is true for some number \(k\) and then prove it for \(k+1\).

Even though our chosen method in the exercise doesn't involve mathematical induction, if we were to apply it, we would need a strong base case, such as the known divisibility of the product of any set of five chosen consecutive integers, and then show that the pattern holds as we move through the number line, ensuring divisibility by 120 consistently.