2017 AMC 10B Problems/Problem 23
Contents
Problem 23
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 only need to find the remainders of N when divided by 5 and 9 to determine the answer. By inspection, . The remainder when is divided by is , but since , we can also write this as , which has a remainder of 0 mod 9. Therefore, by inspection, the answer is .
Note: the sum of the digits of is .
Solution 2
Noting the solution above, we try to find the sum of the digits to figure out its remainder when divided by . From thru , the sum is . thru , the sum is , thru is , and thru is . Thus the sum of the digits is , and thus is divisible by . Now, refer to the above solution. and . From this information, we can conclude that and . Therefore, and so the remainder is
See Also
2017 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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