Question 01 (a)
Show that divides for every positive integer n.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Remember that . An induction argument works or a simple counting argument will suffice.
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
The problem is equivalent to showing that divides . So we must show that contains n copies of both 2 and 3. Now, notice that in we have that contains 2n consecutive numbers of which n are even. Hence . Now, notice that contains 3n consecutive numbers and so n of these must be divisible by 3 (one in every 3 consecutive numbers is divisible by 3. Hence and combining these results (since 2 and 3 are coprime) gives us our result.
For a proof by induction, notice that . Assume that . For , we have
Now, and . Thus, it suffices to show that . Clearly and since this a product of three consecutive numbers, one must be even and so 2 also divides this product. Thus and this completes the proof.