Difference between revisions of "2020 AMC 10B Problems/Problem 7"

Latest revision as of 23:00, 6 February 2024

Problem

How many positive even multiples of $3$ less than $2020$ are perfect squares?

$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 12$

Solution 1

Any even multiple of $3$ is a multiple of $6$ , so we need to find multiples of $6$ that are perfect squares and less than $2020$ . Any solution that we want will be in the form $(6n)^2$ , where $n$ is a positive integer. The smallest possible value is at $n=1$ , and the largest is at $n=7$ (where the expression equals $1764$ ). Therefore, there are a total of $\boxed{\textbf{(A)}\ 7}$ possible numbers.-PCChess

Solution 2

A even multiple square of $3$ can be represented by $3^2 \cdot 2^2 \cdot x^2$ , where $3^2$ is the multiple or $3$ and $2^2$ makes it even. Simplifying we have $36^2 \cdot x^2$ . We can divide $2020$ by $36$ (floor) and get $56$ see the result. We can then see that there are $7$ different values for $x$ . It can't be larger or else $x^2 > 56$ . And thus $\boxed{\textbf{(A) }7}$

~ Wiselion

Solution 3

It can be seen that the problem is just asking for squares that are multiples of six. Thus, all squares of multiples of six can be listed out: $6^2$ , $12^2$ , $18^2$ , $24^2$ , $30^2$ , $36^2$ , and $42^2$ . Since $48^2=2196 > 2020$ , there are $\boxed{\textbf{(A) }7}$ valid answers. ~airbus-a321, November 2023

Video Solution (HOW TO CREATIVELY PROBLEM SOLVE!!!)

https://www.youtube.com/watch?v=igjvQv-TCGE

Check It Out! Short & Straight-Forward Solution ~Education, The Study of Everything

Video Solutions

https://youtu.be/OHR_6U686Qg

https://youtu.be/5cDMRWNrH-U

~savannahsolver

Video Solution by OmegaLearn

https://youtu.be/ZhAZ1oPe5Ds?t=2241

~ pi_is_3.14

@@ Line 1: / Line 1: @@
-Even numbers are divisible by 2, and thus the numbers we are looking for are divisible by 3 X 2, or 6. The square numbers will be 36 X other square numbers. 36X1, 36X4, 36X9, 36X16, 36X25, 36X36, 36X49 are all less than 2020. There are 7 of these square numbers divisible by 6, so the answer is 7
+==Problem==
+How many positive even multiples of <math>3</math> less than <math>2020</math> are perfect squares?
+<math>\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\  9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 12</math>
+==Solution 1==
+Any even multiple of <math>3</math> is a multiple of <math>6</math>, so we need to find multiples of <math>6</math> that are perfect squares and less than <math>2020</math>. Any solution that we want will be in the form <math>(6n)^2</math>, where <math>n</math> is a positive integer. The smallest possible value is at <math>n=1</math>, and the largest is at <math>n=7</math> (where the expression equals <math>1764</math>). Therefore, there are a total of <math>\boxed{\textbf{(A)}\ 7}</math> possible numbers.-PCChess
+==Solution 2==
+A even multiple square of <math>3</math> can be represented by <math>3^2 \cdot 2^2 \cdot x^2</math>, where <math>3^2</math> is the multiple or <math>3</math> and <math>2^2</math> makes it even. Simplifying we have <math>36^2 \cdot x^2</math>. We can divide <math>2020</math> by <math>36</math> (floor) and get <math>56</math> see the result. We can then see that there are <math>7</math> different values for <math>x</math>. It can't be larger or else <math>x^2 > 56</math>. And thus <math>\boxed{\textbf{(A) }7}</math>
+~ Wiselion
+==Solution 3==
+It can be seen that the problem is just asking for squares that are multiples of six. Thus, all squares of multiples of six can be listed out: <math>6^2</math>, <math>12^2</math>, <math>18^2</math>, <math>24^2</math>, <math>30^2</math>, <math>36^2</math>, and <math>42^2</math>. Since <math>48^2=2196 > 2020</math>, there are <math>\boxed{\textbf{(A) }7}</math> valid answers.
+~airbus-a321, November 2023
+==Video Solution (HOW TO CREATIVELY PROBLEM SOLVE!!!)==
+https://www.youtube.com/watch?v=igjvQv-TCGE
+Check It Out! Short & Straight-Forward Solution
+~Education, The Study of Everything
+==Video Solutions==
+https://youtu.be/OHR_6U686Qg
+https://youtu.be/5cDMRWNrH-U
+~savannahsolver
+==Video Solution by OmegaLearn==
+https://youtu.be/ZhAZ1oPe5Ds?t=2241
+~ pi_is_3.14
+==See Also==
+{{AMC10 box|year=2020|ab=B|num-b=6|num-a=8}}
+{{MAA Notice}}

2020 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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