If {x} is the product of all even integers from 1 to x inclusive, what is the greatest prime factor of {22} + {20}?
A) 23
B) 20
C) 11
D) 5
E) 2
OA is a
What is the correct answer here how do you arrive at it? please help me out.
Thanks
Hi Roland2rule,
Let's take a look at your question.
Since, {x} is the product of all even integers from 1 to x inclusive, therefore we can write {22} as,
$$\left\{22\right\}=22\times20\times18\times16\times...\times2$$
Similarly {20} can be written in product form as:
$$\left\{20\right\}=20\times18\times16\times...\times2$$
Now we are asked to find the greatest prime factor of {22}+{20}. For this purpose let's find the sum first.
$$\left\{22\right\}+\left\{20\right\}=\left(22\times20\times18\times16\times...\times2\right)+\left(20\times18\times16\times...\times2\right)$$
Factor out (20 x 18 x 16 x .. x 2), we get
$$\left\{22\right\}+\left\{20\right\}=\left(20\times18\times16\times...\times2\right)\left(22+1\right)$$
$$\left\{22\right\}+\left\{20\right\}=\left(20\times18\times16\times...\times2\right)23$$
Now we can see all the factors of {22}+{20}, Except 23 which is a prime factor, all other factors are even integers that are not prime.
Therefore, the greatest prime factor of {22}+{20} is 23.
Option A is correct.
Hope it helps.
I am available if you'd like any follow up.
