What! This Easy!? 2

Number Theory Level 2

What is the unit's digit of

0 ! + 1 ! + 2 ! + 3 ! + 4 ! + 5 ! + 6 ! + . . . m ! 0! + 1! + 2! +3! + 4! + 5! + 6! + ... m!

6 4 3 9

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Math Man by math man , 20, Indonesia

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2 solutions

Curtis Clement
Jan 29, 2015

0 ! + 1 ! + 2 ! + 3 ! + 4 ! = 1 + 1 + 2 + 6 + 24 = 34 ≡ 4 m o d ( 10 ) 0! +1! +2! +3!+4! = 1+1+2+ 6+24 = 34\equiv4\mod(10) . N o w 10 ∣ n ! i f n ≥ 5 Now\ 10|n!\ if \ n\geq5 ∴ 0 ! + 1 ! + 2 ! + 3 ! + 4 ! + 5 ! + 6 ! + . . . h a s a u n i t ′ s d i g i t o f 4 \therefore\ 0! +1! + 2! + 3! + 4! + 5! + 6!+... has \ a \ unit's \ digit \ of \ 4

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Lu Chee Ket
Jan 29, 2015

874 + 7 ! + 8 ! + ... m! = 874 + 5040 + 40320 + ... m!

0! + 1! + 2! + 3! + 4! = 34, from 5! on wards, all shall have at least one zero.

Unit 4 shall not change in sum when continued.

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