Difference between revisions of "2023 AMC 8 Problems/Problem 4"

(Solution 1)
(Problem)
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The numbers from 1 to 49 are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number 7. How many of these four numbers are prime?
 
The numbers from 1 to 49 are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number 7. How many of these four numbers are prime?
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[asy]
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size(6cm);
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fill((4,0)--(5,0)--(5,1)--(4,1)--cycle,mediumgray);
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fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,mediumgray);
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fill((1,3)--(1,4)--(2,4)--(2,3)--cycle,mediumgray);
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fill((0,4)--(0,5)--(1,5)--(1,4)--cycle,mediumgray);
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 +
label(scale(.9)*"<math>1</math>", (3.5,3.5));
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label(scale(.9)*"<math>2</math>", (4.5,3.5));
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label(scale(.9)*"<math>3</math>", (4.5,4.5));
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label(scale(.9)*"<math>4</math>", (3.5,4.5));
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label(scale(.9)*"<math>5</math>", (2.5,4.5));
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label(scale(.9)*"<math>6</math>", (2.5,3.5));
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label(scale(.9)*"<math>7</math>", (2.5,2.5));
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 +
draw((1,0)--(1,7)--(2,7)--(2,0)--(3,0)--(3,7)--(4,7)--(4,0)--(5,0)--(5,7)--(6,7)--(6,0)--(7,0)--(7,7),gray);
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 +
draw((0,1)--(7,1)--(7,2)--(0,2)--(0,3)--(7,3)--(7,4)--(0,4)--(0,5)--(7,5)--(7,6)--(0,6)--(0,7)--(7,7),gray);
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 +
draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(1.25));
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[/asy]
  
 
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4</math>
 
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4</math>

Revision as of 12:09, 25 January 2023

Problem

The numbers from 1 to 49 are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number 7. How many of these four numbers are prime?

[asy] size(6cm); fill((4,0)--(5,0)--(5,1)--(4,1)--cycle,mediumgray); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,mediumgray); fill((1,3)--(1,4)--(2,4)--(2,3)--cycle,mediumgray); fill((0,4)--(0,5)--(1,5)--(1,4)--cycle,mediumgray);

label(scale(.9)*"$1$", (3.5,3.5)); label(scale(.9)*"$2$", (4.5,3.5)); label(scale(.9)*"$3$", (4.5,4.5)); label(scale(.9)*"$4$", (3.5,4.5)); label(scale(.9)*"$5$", (2.5,4.5)); label(scale(.9)*"$6$", (2.5,3.5)); label(scale(.9)*"$7$", (2.5,2.5));

draw((1,0)--(1,7)--(2,7)--(2,0)--(3,0)--(3,7)--(4,7)--(4,0)--(5,0)--(5,7)--(6,7)--(6,0)--(7,0)--(7,7),gray);

draw((0,1)--(7,1)--(7,2)--(0,2)--(0,3)--(7,3)--(7,4)--(0,4)--(0,5)--(7,5)--(7,6)--(0,6)--(0,7)--(7,7),gray);

draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(1.25)); [/asy]

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Solution 1

First, we fill out the entire grid. We find that the $4$ numbers are $39,19,23,47$. The numbers $19,23,$ and $47$ are prime, so there are $\boxed{\textbf{(D) }3}$ prime numbers.

~MathFun1000 (minor edits apex304 and ILoveMath31415926535)

Solution 2

Fill out the entire grid to count that there are $\boxed{\text{(D)}3}$ prime numbers -apex304

Video Solution by Magic Square

https://youtu.be/-N46BeEKaCQ?t=5392

See Also

2023 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
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 AJHSME/AMC 8 Problems and Solutions

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