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Factorial n, written n!, is defined by: n! = 1 × 2 × 3 × · · · × n.
What is the remainder when 1!+2!+3!+4!+5!+6!+7!+8!+9!+10! is divided by 5?
A. 0 B. 1 C. 2 D. 3 E. 4
D. 3
We have
1! = 1,
2! = 1 × 2 = 2,
3! = 1 × 2 × 3 = 6,
4! = 1 × 2 × 3 × 4 = 24,
5! = 1 × 2 × 3 × 4 × 5 = 120.
We see that 5! has 0 as its units digit.
Similarly, for n > 5, n! = 1 × 2 × 3 × 4 × 5 × · · · × n and is therefore a multiple of 10.
Hence n! also has units digit 0 for n > 5. It follows that the units digit of 1! + 2! + 3! + 4! + 5! + 6! + 7! + 8! + 9! + 10! is the same as the units digit of 1! + 2! + 3! + 4!.
Now 1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33.
Hence its units digit is 3.
Therefore the units digit of 1! + 2! + 3! + 4! + 5! + 6! + 7! + 8! + 9! + 10! is also 3. Hence theremainder when this number is divided by 5 is 3
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