Stuck with factorial exercise

GRE Quant Problem Solving
1

I am pretty stuck with the factorial part. This topic has two parts (please bear with me) - confirmation of my method in solving a factorial and question from prep swift

Part 1 - confirmation of technique in solving a factorial

Sample question: solve a factorial of 8!

8! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8

Step 1: I would just pair first and last digit for ease of pairing solution

(1 x 8) x (2 x 7) x (3 x 6) x (4 x 5) = 8 x 14 x 18 x 20

Step 2: use calculator to solve it leading to 40,320

Part 2 - Unsure how to solve this question from Prep Swift

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The proposed solution is as follows

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I am really lost with the explanation of the proposed solution. I need someone to help breaking it down further for me as I can’t follow the reasoning.

My original method in trying to identify the factor is as follows:

10!

10 / 2 = 5 (note: divide by the smallest prime first which is 2)
5 / 5 = 1 (note: the next factorisation prime is 5)

Therefore, the factors are 2^1 and 5^1 or just 2 and 5. But clearly my method is wrong.

2

Part 2 Updates - identifying factor of 10!

Correct me if I am wrong but any factorial number from 1!, 2!, 3!..9!, 10! is considered as a factor of 10!

Likewise, any integer that appears from 1 to 10 including those that are factors of a number are considered as a factor of 10!

For example, 26 is not a factor because there’s no number from 1 to 10 when multiplies will give 26

27 is a factor because 9 x 3 = 27 and 9 and 3 are smaller than 10

34 is not a factor because there’s no number from 1 to 10 when multiplies will give 34

120 is a factor because 2 x 6 x 10 are within the sequence of 1 to 10.

Is my understanding correct?

3

Yes, 34 is not a factor because 10! doesn’t have 17.
26 is not because 10! doesn’t have 13.
Likewise for others also.
Hope this helps.

4

Many many thanks for taking time to respond to my queries. I really appreciate it!

Just one last one - whether my method for part 1 is correct? Thank again for taking time! :pray: