Difference between revisions of "2023 AMC 12A Problems/Problem 3"

Revision as of 19:31, 10 June 2024

The following problem is from both the 2023 AMC 10A #3 and 2023 AMC 12A #3, so both problems redirect to this page.

1 Problem
2 Solution 2 (slightly refined)
3 Solution 3 (the best)
4 Solution 4
5 Solution 5 (DO NOT DO ON AN ACTUAL TEST)
6 Video Solution (easy to understand) by Power Solve
7 Video Solution by Math-X (First understand the problem!!!)
8 Video Solution by CosineMethod [🔥Fast and Easy🔥]
9 Video Solution
10 Video Solution (🚀 Just 2 min 🚀)
11 Video Solution (easy to digest) by Power Solve
12 Video Solution (Easy to Understand) by DR.GOOGLE (YT: Pablo's Math)
13 See Also

Problem

How many positive perfect squares less than $2023$ are divisible by $5$ ?

$\textbf{(A) } 8 \qquad\textbf{(B) }9 \qquad\textbf{(C) }10 \qquad\textbf{(D) }11 \qquad\textbf{(E) } 12$

Solution 2 (slightly refined)

Since $\left \lfloor{\sqrt{2023}}\right \rfloor = 44$ , there are $\left \lfloor{\frac{44}{5}}\right \rfloor = \boxed{\textbf{(A) 8}}$ perfect squares less than 2023.

~not_slay

Solution 3 (the best)

Since $5$ is prime, each solution must be divisible by $5^2=25$ . We take $\left \lfloor{\frac{2023}{25}}\right \rfloor = 80$ and see that there are $\boxed{\textbf{(A) 8}}$ positive perfect squares no greater than $80$ .

~jwseph

Solution 4

We know the highest value would be at least $40$ but less than $50$ so we check $45$ , prime factorizing 45. We get $3^2 \cdot 5$ . We square this and get $81 \cdot 25$ . We know that $80 \cdot 25 = 2000$ , then we add 25 and get $2025$ , which does not satisfy our requirement of having the square less than $2023$ . The largest multiple of $5$ that satisfies this is $40$ and the smallest multiple of $5$ that works is $5$ so all multiples of $5$ from $5$ to $40$ satisfy the requirements. Now we divide each element of the set by $5$ and get $1-8$ so there are $\boxed{\textbf{(A) 8}}$ solutions.

~kyogrexu (minor edits by vadava_lx)

Solution 5 (DO NOT DO ON AN ACTUAL TEST)

Since the perfect squares have to be divisible by 5, then we know it has to be 5 times some number squared (5*x)^2. With this information, you can fique out every single product of 5 and another number squared to count how many perfect squares are divisible by 5 that are less than 2023. (EX: 5^2 = 25, 10^2 = 100, ... 40^2 = 1600) With this you get a max of 40^2, or 8 solutions. PLEASE DO NOT do this problem this way, it takes way too much time.

~BlueShardow

Video Solution (easy to understand) by Power Solve

https://youtu.be/YXIH3UbLqK8?si=aIYHWEU82uUu21fQ&t=165

Video Solution by Math-X (First understand the problem!!!)

https://youtu.be/cMgngeSmFCY?si=E0a8wvcNRoeg2A3X&t=422

Video Solution by CosineMethod [🔥Fast and Easy🔥]

https://www.youtube.com/watch?v=wNH6O8D-7dY

Video Solution

https://youtu.be/w7RBPIatRNE

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution (🚀 Just 2 min 🚀)

https://youtu.be/Z3fmCkuHG3c

~Education, the Study of Everything

Video Solution (easy to digest) by Power Solve

https://www.youtube.com/watch?v=8huvzWTtgaU

Video Solution (Easy to Understand) by DR.GOOGLE (YT: Pablo's Math)

https://youtu.be/BNhRdnOu-jI

@@ Line 26: / Line 26: @@
 ~BlueShardow
+==Video Solution (easy to understand) by Power Solve==
+https://youtu.be/YXIH3UbLqK8?si=aIYHWEU82uUu21fQ&t=165
 ==Video Solution by Math-X (First understand the problem!!!)==

2023 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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

2023 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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 12 Problems and Solutions

Page

Toolbox

Search