2017 AMC 12B Problems/Problem 19
Problem
Let be the
-digit number that is formed by writing the integers from
to
in order, one after the other. What is the remainder when
is divided by
?
Solution
We will consider this number and
. By looking at the last digit, it is obvious that the number is
. To calculate the number
, note that
so it is equivalent to
Thus it is and
, so it is
since 0 mod 9 is congruent to 9 mod 9, we multiply both the coefficient and the modulus to get 36 mod 45 which is congruent to 9 mod 45
See Also
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
2017 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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