Difference between revisions of "2017 AMC 12B Problems/Problem 19"

Latest revision as of 16:06, 2 September 2024

Problem

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 1

We will consider this number $\bmod\ 5$ and $\bmod\ 9$ . By looking at the last digit, it is obvious that the number is $\equiv 4\bmod\ 5$ . To calculate the number $\bmod\ 9$ , note that

$123456\cdots 4344 \equiv 1+2+3+4+5+6+7+8+9+(1+0)+(1+1)+\cdots+(4+3)+(4+4) \equiv 1+2+\cdots+44 \bmod\ 9,$

so it is equivalent to

$\frac{44\cdot 45}{2} = 22\cdot 45 \equiv 0\bmod\ 9.$

Let $x$ be the remainder when this number is divided by $45$ . We know that $x\equiv 0 \pmod {9}$ and $x\equiv 4 \pmod {5}$ , so by the Chinese remainder theorem, since $9(-1)\equiv 1 \pmod{5}$ , $x\equiv 5(0)+9(-1)(4) \pmod {5\cdot 9}$ , or $x\equiv -36 \equiv 9 \pmod {45}$ . So the answer is $\boxed {\textbf {(C)}}$ .

Solution 2

We know that this number is divisible by $9$ because the sum of the digits is $270$ , which is divisible by $9$ . If we subtracted $9$ from the integer we would get $1234 \cdots 4335$ , which is also divisible by $5$ and by $45$ . Thus the remainder is $9$ , or $\boxed{\textbf{C}}$ .

Solution 3 (Beginner's Method)

To find the sum of digits of our number, we break it up into $5$ cases, starting with $0$ , $1$ , $2$ , $3$ , or $4$ .

Case 1: $1+2+3+\cdots+9 = 45$ ,

Case 2: $1+0+1+1+1+2+\cdots+1+8+1+9 = 55$ (We add 10 to the previous cases, as we are in the next ten's place)

Case 3: $2+0+2+1+\cdots+2+9 = 65$ ,

Case 4: $3+0+3+1+\cdots+3+9 = 75$ ,

Case 5: $4+0+4+1+\cdots+4+4 = 30$ ,

Thus the sum of the digits is $45+55+65+75+30 = 270$ , so the number is divisible by $9$ . We notice that the number ends in " $4$ ", which is $9$ more than a multiple of $5$ . Thus if we subtracted $9$ from our number it would be divisible by $5$ , and $5\cdot 9 = 45$ . (Multiple of n - n = Multiple of n)

So our remainder is $\boxed{\textbf{(C)}\,9}$ , the value we need to add to the multiple of $45$ to get to our number.

Solution 4

We notice that $10^{k}\equiv 10 \pmod {45}$ .

Hence $N = 44+43\cdot10^{2}+42\cdot10^{4}+\cdots+10^{78} \equiv 44+10\cdot(1+2+3+\cdots+43)\equiv 9 \pmod {45}.$

Choose $\boxed{\textbf{(C)}\,9}$

~ PythZhou

Solution 5 (Solution 2 but more in-depth)

The sum of all of the digits is just the sum of consecutive numbers from $1 -> 44$ or $\frac{44}{2}(45) = 990$ . The prime factorization of $45$ is $3^2 * 5$ . So if a number is divisible by $45$ it has to both be divisible by $9$ and $5$ . The first number that satisfies this ends in $35$ because the ten's digit is one greater than the ten's digit in the consecutive numbers from $1$ to $44$ , and the one's digit is one less than this number, and the number ends in a 5. The difference between $35$ and $44$ is $9$ giving $\boxed{\textbf{(C)}\,9}$ .

~PeterDoesPhysics

Video Solution by OmegaLearn

https://youtu.be/zfChnbMGLVQ?t=3342

~ pi_is_3.14

@@ Line 49: / Line 49: @@
 ~ PythZhou
+==Solution 5 (Solution 2 but more in-depth)==
+The sum of all of the digits is just the sum of consecutive numbers from <math>1 -> 44</math> or <math>\frac{44}{2}(45) = 990</math>. The prime factorization of <math>45</math> is <math>3^2 * 5</math>. So if a number is divisible by <math>45</math> it has to both be divisible by <math>9</math> and <math>5</math>. The first number that satisfies this ends in <math>35</math> because the ten's digit is one greater than the ten's digit in the consecutive numbers from <math>1</math> to <math>44</math>, and the one's digit is one less than this number, and the number ends in a 5. The difference between <math>35</math> and <math>44</math> is <math>9</math> giving <math>\boxed{\textbf{(C)}\,9}</math>.
+~PeterDoesPhysics
 == Video Solution by OmegaLearn==

2017 AMC 12B (Problems • Answer Key • Resources)
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