What is the largest prime factor of the product of all even numbers from 2 through 200?

Question

I don't know how to solve this. I tried multiplying all of the even numbers from 2 through 200, but the number got too big. Possible answers:

A) 47

B) 97

C) 199

D) 2019

Answer 1

Look at it this way:

You have: $$(2\times 1) \times(2\times2)\times(2\times3)\times\ldots\times(2\times100)$$ $$=2^{100}\times(1\times 2\times 3\times\ldots \times 100)$$

What is the largest single element of this factorisation which is prime? $(100$ is the largest single element, but it's not prime, so work backwards from there$)$

Answer 2

By Euclid's lemma, any prime dividing a product must divide one of the factors of the product. So you're looking for a prime dividing one of the even numbers from $2$ through $200$. That means you're looking for a prime dividing one of the numbers from $1$ through $100$ (since there are primes larger than $2$ dividing the product). The largest one is therefore $97$, since $98, 99, $ and $100$ are not prime.

Answer 3

Note that $$\prod_{k=1}^{n} 2k=2^{n}\cdot n!$$

Hence the problem is: $$\prod_{k=1}^{100} 2k=2^{100}\cdot 100!$$

It's clear to see that the biggest prime factor is $97.$