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2017 AMC 10B Problems/Problem 23

Revision as of 15:43, 17 January 2020 by Aops turtle (talk | contribs) (→‎Solution)

Problem 23

Let $N=123456789101112\dots4344$ be the $79$-digit number that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 44$

Solution

We only need to find the remainders of N when divided by 5 and 9 to determine the answer. By inspection, $N \equiv 4 \text{ (mod 5)}$. The remainder when $N$ is divided by $9$ is $1+2+3+4+ \cdots +1+0+1+1 +1+2 +\cdots + 4+3+4+4$, but since $10 \equiv 1 \text{ (mod 9)}$, we can also write this as $1+2+3 +\cdots +10+11+12+ \cdots 43 + 44 = \frac{44 \cdot 45}2 = 22 \cdot 45$, which has a remainder of 0 mod 9. Therefore, the answer is $\boxed{\textbf{(C) } 9}$.

Solution 2

Noting the solution above, we try to find the sum of the digits to figure out its remainder when divided by $9$. From $1$ thru $9$, the sum is $45$. $10$ thru $19$, the sum is $55$, $20$ thru $29$ is $65$, and $30$ thru $39$ is $75$. Thus the sum of the digits is $45+55+65+75+4+5+6+7+8 = 240+30 = 270$, and thus $N$ is divisible by $9$. Now, refer to the above solution. $N \equiv 4 \text{ (mod 5)}$ and $N \equiv 0 \text{ (mod 9)}$. From this information, we can conclude that $N \equiv 54 \text{ (mod 5)}$ and $N \equiv 54 \text{ (mod 9)}$. Therefore, $N \equiv 54 \text{ (mod 45)}$ and $N \equiv 9 \text{ (mod 45)}$ so the remainder is $\boxed{\textbf{(C) }9}$

Solution 3

Because a number is equivalent to the sum of its digits modulo 9, we have that $N\equiv 1+2+3+4+5+...+44\equiv \frac{44\times 45}{2}\equiv 0\pmod{9}$. Furthermore, we see that $N-9$ ends in the digit 5 and thus is divisible by 5, so $N-9$ is divisible by 45, meaning the remainder is $\boxed{\textbf{(C) }9}$ -Stormersyle

See Also

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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