prove that 24∣a(a2−1)
Solution
Sure, let's prove that 24 divides a(a^2 - 1).
Step 1: Rewrite the expression First, we can rewrite the expression a(a^2 - 1) as a(a - 1)(a + 1). This is because a^2 - 1 is a difference of squares, which can be factored into (a - 1)(a + 1).
Step 2: Consider the properties of consecutive integers The numbers a - 1, a, and a + 1 are three consecutive integers.
Step 3: Apply the divisibility rule One of the properties of consecutive integers is that at least one of them is divisible by 2, and at least one of them is divisible by 3.
Step 4: Prove divisibility by 24 Since 24 = 2^3 * 3, and we know that one of our integers is divisible by 2 and another is divisible by 3, we can say that the product of these three consecutive integers (a - 1, a, a + 1) is divisible by 24.
Therefore, we have proved that 24 divides a(a^2 - 1).
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