Just Factorials around

Number Theory Level 3

S n = 1 ! + 2 ! + 3 ! + … + n ! \large S_n = 1! + 2! + 3! + \ldots + n!

Suppose for integer n > 3 n>3 , we define S n S_n as above. If a n a_n is the maximum positive integer that satisfy the equation S n 24 = a n + λ 24 \frac{S_n}{24} = a_n + \frac{ \lambda}{24} for constant λ \lambda , find the value of λ \lambda .


The answer is 9.

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Tanishq Varshney by Tanishq Varshney , 23, India

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2 solutions

Tanishq Varshney
Jun 10, 2015

S n = 1 ! + 2 ! + 3 ! + ( 4 ! + 5 ! + . . . + n ! ) S_{n}=1!+2!+3!+(4!+5!+...+n!)

Taking 4 ! 4! common

S n = 9 + 24 ( 1 + 5 + 6 × 5 + . . . . ) S_{n}=9+24(1+5+6\times 5+....)

S n 24 = ( 1 + 5 + 30 + . . . ) + 9 24 \huge{\frac{S_{n}}{24}=(1+5+30+...)+\frac{9}{24}}

S n 24 = a n + 9 24 \huge{\frac{S_{n}}{24}=a_n+\frac{9}{24}}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

It's better to state that 24 24 divides 4 ! , 5 ! , … , n ! 4!, 5!, \ldots , n! .

Bonus question : Can you generalize this problem?

Mahmoud Sayed
Jun 12, 2015

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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