+0  
 
0
101
2
avatar+739 

The least common multiple of 1!+2!, 2!+3!, 3!+4!, 4!+5!, 5!+6!, 6!+7!, 7!+8!, and 8!+9! can be expressed in the form \(a\cdot b!\), where a and b are integers and b is as large as possible. What is a+b?

MIRB16  Apr 28, 2018
 #1
avatar+942 
+2

We can factor each of these numbers.

 

1! + 2! = 1! ( 1 + 2 ) 

 

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

 

3! + 4! = 3! ( 1 + 4 )

 

4! + 5! = 4! ( 1 + 5 )

 

5! + 6! = 5! ( 1 + 6 )

 

6! + 7! = 6! ( 1 + 7 )

 

7! + 8! = 7! ( 1 + 8 )

 

8! + 9! = 8! ( 1 + 9 )

 

Once again:

 

1! + 2! = 1! ( 3 ) 

 

2! + 3! = 2! ( 4 )

 

3! + 4! = 3! ( 5 )

 

4! + 5! = 4! ( 6 )

 

5! + 6! = 5! ( 7 )

 

6! + 7! = 6! ( 8 )

 

7! + 8! = 7! ( 9 )

 

8! + 9! = 8! ( 10 )

 

We can split the number into two parts, a! * b

 

The LCM of 1!, 2!, 3!, ... , 8! is 8!.

 

The LCM of 3, 4, 5, ... , 10 is 2520

 

The LCM is 2520 * 8! 

 

a = 2520

b = 8

 

a + b = 2528

 

I hope this helped,

 

Gavin

GYanggg  Apr 28, 2018
 #2
avatar
-1

The lcm of[1!+2!, 2!+3!, 3!+4!, 4!+5!, 5!+6!, 6!+7!, 7!+8!, 8!+9! ] =10! =3,628,800.

3,628,800 =10 x 9! =So, a + b = 10 + 9 =19. OR:

Find the least common multiple:
lcm(3, 8, 30, 144, 840, 5760, 45360, 403200)

Find the prime factorization of each integer:
The prime factorization of 3 is:
3 = 3^1

The prime factorization of 8 is:
8 = 2^3

The prime factorization of 30 is:
30 = 2×3×5

The prime factorization of 144 is:
144 = 2^4×3^2

The prime factorization of 840 is:
840 = 2^3×3×5×7

The prime factorization of 5760 is:
5760 = 2^7×3^2×5

The prime factorization of 45360 is:
45360 = 2^4×3^4×5×7

The prime factorization of 403200 is:
403200 = 2^8×3^2×5^2×7
Find the largest power of each prime factor.
The largest power of 2 that appears in the prime factorizations is 2^8.
The largest power of 3 that appears in the prime factorizations is 3^4.
The largest power of 5 that appears in the prime factorizations is 5^2.
The largest power of 7 that appears in the prime factorizations is 7^1.
Therefore lcm(3, 8, 30, 144, 840, 5760, 45360, 403200) = 2^8×3^4×5^2×7^1 = 3628800:

 lcm(3, 8, 30, 144, 840, 5760, 45360, 403200) = 3628800

The LCM of {3, 8, 30, 144, 840, 5,760, 45,360, 403,200} =3,628,800 = 10!. As above, 10! can be written as: 1 x 10! =a + b! =1 + 10 =11, which is probably the more accurate answer, since the question wants "b" to be "as large as possible".

[Courtesy of Mathematica 11 Home Edition]

Guest Apr 28, 2018
edited by Guest  Apr 28, 2018
edited by Guest  Apr 28, 2018
edited by Guest  Apr 28, 2018

11 Online Users

avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.