2020 AMC 10B Problems/Problem 7
Contents
Problem
How many positive even multiples of less than
are perfect squares?
Solution 1
Any even multiple of is a multiple of
, so we need to find multiples of
that are perfect squares and less than
. Any solution that we want will be in the form
, where
is a positive integer. The smallest possible value is at
, and the largest is at
(where the expression equals
). Therefore, there are a total of
possible numbers.-PCChess
Solution 2
A even multiple square of can be represented by
, where
is the multiple of
and
makes it even. Simplifying we have
. We can divide
by
(floor) and get
as the result. We can then see that there are
different values for
. It cannot be larger than
or else
. And thus
~ Wiselion
~<B+
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: ,
,
,
,
,
, and
. Since
, there are
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
~savannahsolver
Video Solution by OmegaLearn
https://youtu.be/ZhAZ1oPe5Ds?t=2241
~ pi_is_3.14
See Also
2020 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.