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# Divisibility Practice

Prove that $$(2014!)!$$ is divisible by $$((2014 \cdot 2013)!)^{2012!}$$

Note by Pratik Shastri
2 years, 10 months ago

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Let us have $$2014!$$ balls. Let us divide them in $$2012!$$ groups, each group having $$2013.2014$$ balls.

Let's assume that balls of each group are of different colour, but all balls of the same group are the of the same colour.

Number of ways of doing so is, $\dfrac{(2014!)!}{((2014.2013)!)^{2012!}}$ Number of ways of doing something is always a positive integer. Hence Proved. $$\boxed{.}$$

- 2 years, 10 months ago

Using Combinatorics for a Number Theory Problem is simply amazing!

- 2 years, 10 months ago

Excellent! +1

- 2 years, 10 months ago

@Satvik Golechha this is great method see solution of @John Ashley Capellan here

- 2 years, 10 months ago

superb way to do this......

- 2 years, 9 months ago