What's my Last Digit?

Number Theory Level 4

⌊ 10 2000 10 100 + 3 ⌋ \left \lfloor \frac { { 10 }^{ 2000 } }{ { 10 }^{ 100 }+3 } \right\rfloor

Find the unit digit of the expression above.


The answer is 3.

Ellen Shang by Ellen Shang , 23, Canada

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2 solutions

Patrick Corn
Dec 15, 2014

Let x = 1 0 100 x = 10^{100} . Then x 20 x + 3 = x 19 − 3 x 18 + 9 x 17 + ⋯ + 3 18 x − 3 19 + 3 20 x + 3 \frac{x^{20}}{x+3} = x^{19} - 3x^{18} + 9x^{17} + \cdots + 3^{18}x - 3^{19} + \frac{3^{20}}{x+3} That last fraction is between 0 0 and 1 1 , and the positive powers of x x are all divisible by 10 10 , so the units digit is − 3 19 -3^{19} mod 10 10 , which is 3 \fbox{3} .

8 Helpful 0 Interesting 0 Brilliant 0 Confused

Exactly nicely done

U Z - 6 years, 5 months ago

Exactly same solution. Good job.

Kunal Verma - 6 years, 2 months ago

(10^2000/(10^100+3)) mod 10 =3

0 Helpful 0 Interesting 0 Brilliant 0 Confused

How do you get the answer of that fractional modulo operation?

Antonio Hugo - 6 years, 6 months ago

Can you explain how you arrived at that answer?

Calvin Lin Staff - 6 years, 6 months ago

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