# FACTORING WITH FACTORIALS

Write as a product using factorization:

Problem 1 :

5! + 4!

Solution :

5! + 4!

5! Can be written as 5 × 4!

= (5 × 4!) + 4!

Factoring 4!, we get

= 4! (5 + 1)

= 4! (6)

5! + 4! = 6 × 4!

Problem 2 :

11! - 10!

Solution :

11! - 10!

11! Can be written as 11 × 10!

= (11 × 10!) - 10!

Factoring 10!, we get

= 10! (11 - 1)

= 10! (10)

11! - 10! = 10 × 10!

Problem 3 :

6! + 8!

Solution :

6! + 8!

8! Can be written as 8 × 7 × 6!

= 6! + (8 × 7 × 6!)

Factoring 6!, we get

= 6! (1 + 56)

6! + 8! = 57 × 6!

Problem 4 :

12! - 10!

Solution :

12! - 10!

12! Can be written as 12 × 11 × 10!

= (12 × 11 × 10!) - 10!

Factoring 10!, we get

= 10! (12 × 11) - 1

= 10! (132 - 1)

= 10! (131)

12! - 10! = 131 × 10!

Problem 5 :

9! + 8! + 7!

Solution :

9! + 8! + 7!

9! Can be written as 9 × 8 × 7!

= (9 × 8 × 7!) + 8! + 7!

Factoring 7!, we get

= 7! (9 × 8) + 8 + 1

= 7! (72) + 8 + 1

= 7! × 81

9! + 8! + 7! = 81 × 7!

Problem 6 :

7! - 6! + 8!

Solution :

7! - 6! + 8!

8! Can be written as 8 × 7 × 6!

= 7! - 6! + (8 × 7 × 6!)

Factoring 7!, we get

= 6! (8 × 7) + 7 - 1

= 6! (56 + 6)

= 6! (62)

7! - 6! + 8! = 62 × 6!

Problem 7 :

12! - 2 ×11!

Solution :

12! - 2 ×11!

12! Can be written as 12 × 11!

= (12 × 11!) - 2 × 11!

Factoring 11!, we get

= 11! (12) - 2 × 1

= 11! (12 - 2)

= 11! (10)

12! - 2 ×11! = 10 × 11!

Problem 8 :

3 × 9! + 5 × 8!

Solution :

3 × 9! + 5 × 8!

9! Can be written as 9 × 8!

= 3 × (9 × 8!) + 5 × 8!

Factoring 7!, we get

= 8! (3 × 9) + 5(1)

= 8! (27) + 5

= 8! (32)

3 × 9! + 5 × 8! = 32 × 8!

Simplify using factorization :

Problem 1 :

(12! - 11!) / 11

Solution :

(12! - 11!) / 11

12! can be written as 12 × 11!

= [(12 × 11!) - 11!] / 11

Factoring 11!, we get

= 11! (12 - 1) / 11

= 11! (11) / 11

= 11!

Problem 2 :

(10! + 9!) / 11

Solution :

(10! + 9!) / 11

10! Can be written as 10 × 9!

= (10 × 9!) + 9! / 11

Factoring 9!, we get

= [9! (10) + 1] / 11

= 9! (11) / 11

= 9!

Problem 3 :

(10! - 8!) / 89

Solution :

(10! - 8!) / 89

10! Can be written as 10 × 9 × 8!

= (10 × 9 × 8!) - 8! / 89

Factoring 8!, we get

= 8! (10 × 9) - 1 / 89

= 8! (90) - 1 / 89

= 8! (89) / 89

= 8!

Problem 4 :

(10! - 9!) / 9!

Solution :

10! - 9! / 9!

10! Can be written as 10 × 9!

= (10 × 9!) - 9! / 9!

Factoring 9!, we get

= 9! (10 - 1) / 9!

= 9

Problem 5 :

(6! + 5! - 4!) / 4!

Solution :

(6! + 5! - 4!) / 4!

6! Can be written as 6 × 5 × 4!

= [(6 × 5 × 4!) + (5 × 4!) - 4!] / 4!

Factoring 4!, we get

= 4![(6 × 5) + 5 - 1] / 4!

= 4! (30 + 4) / 4!

= 34

Problem 6 :

(n! + (n - 1)!) / (n - 1)!

Solution :

(n! + (n - 1)!) / (n - 1)!

n! can be written as n (n - 1)!

= n (n - 1)! + (n - 1)! / (n - 1)!

Factoring (n - 1)!, we get

= (n - 1)! [(n + 1)] / (n - 1)!

= n + 1

Problem 7 :

(n! - (n - 1)! )/ n - 1

Solution :

(n! - (n - 1)!) / (n - 1)

n! can be written as n(n - 1)

= (n(n - 1)! - (n - 1)!) / (n - 1)

Factoring (n - 1)!, we get

= (n - 1)!(n - 1) / (n - 1)

= (n - 1)!

Problem 8 :

(n + 2)! + [(n + 1)! / (n + 3)]

Solution :

(n + 2)! + [(n + 1)! / (n + 3)]

(n + 2)! can be written as (n + 2) (n + 1)!

= (n + 2) (n + 1)! + (n + 1)! / (n + 3)

Then (n + 1)! In common, so will get

= (n + 1)! [(n + 2) + 1] / (n + 3)

= (n + 1)! [(n + 3)] / (n + 3)

= (n + 1)!

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