Prime factors/number properties

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BTGmoderatorRO: Moderator; Posts: 772; Joined: Wed Aug 30, 2017 6:29 pmFollowed by: 6 members

by BTGmoderatorRO » Sat Oct 28, 2017 3:49 am

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

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Source: Beat The GMAT — Problem Solving | Sat Oct 28, 2017 3:49 am

EconomistGMATTutor: GMAT Instructor; Posts: 555; Joined: Wed Oct 04, 2017 4:18 pmFollowed by: 12 members; Thanked: 180 times

by EconomistGMATTutor » Sat Oct 28, 2017 1:54 pm

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.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8157; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Tue Nov 12, 2019 7:15 pm

BTGmoderatorRO wrote: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

{22} = 22 x 20 x 18 x 16 x ... 4 x 2

{20} = 20 x 18 x 16 x ... 4 x 2

{22} + {20} = (22 x 20 x 18 x 16 x ... 4 x 2) + (20 x 18 x 16 x ... 4 x 2)

From the two terms above, we pull out the common factor of (20 x 18 x 16 x ... 4 x 2), yielding:

{22} + {20} = 20 x 18 x 16 x ... 4 x 2 x (22 + 1)

{22} + {20} = 20 x 18 x 16 x ... 4 x 2 x 23

We see that the greatest prime factor is 23.

Answer: A

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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swerve: Legendary Member; Posts: 2548; Joined: Sun Oct 29, 2017 2:04 pmFollowed by: 6 members

by swerve » Thu Nov 14, 2019 6:58 am

BTGmoderatorRO wrote: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

Now if $\{x\}$ is the product of all even integers from 1 to $x$ inclusive, this means

we can take 20 common from $\{22\} + \{20\}$

$20\times 18 \cdots 2 (22 + 1)$

$23 \times $ Some expression

23 $\Rightarrow$ __A__.

Greatest prime factor.

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