2024 AMC 10A Problems/Problem 1
- The following problem is from both the 2024 AMC 10A #1 and 2024 AMC 12A #1, so both problems redirect to this page.
Contents
[hide]- 1 Problem
- 2 Solution 0 (Direct Computation)
- 3 Solution 1 (Distributive Property but faster)
- 4 Solution 2 (Distributive Property)
- 5 Solution 3 (Solution 1 but Distributive)
- 6 Solution 4 (Modular Arithmetic)
- 7 Solution 5 (Process of Elimination)
- 8 Solution 6 (Faster Distribution)
- 9 Solution 7 (Cubes)
- 10 Solution 8 (Super Fast)
- 11 Video Solution (⚡️ 1 min solve ⚡️)
- 12 Video Solution by Pi Academy
- 13 Video Solution by FrankTutor
- 14 Video Solution Daily Dose of Math
- 15 Video Solution 1 by Power Solve
- 16 Video Solution by SpreadTheMathLove
- 17 Video Solution by Math from my desk
Problem
What is the value of
Solution 0 (Direct Computation)
The likely fastest method will be direct computation. evaluates to
and
evaluates to
. The difference is
Solution by juwushu.
Solution 1 (Distributive Property but faster)
~sorebrek
Solution 2 (Distributive Property)
We have
~MRENTHUSIASM
Solution 3 (Solution 1 but Distributive)
Note that and
, therefore the answer is
.
~Tacos_are_yummy_1
Solution 4 (Modular Arithmetic)
Evaluating the given expression yields
, so the answer is either
or
. Evaluating
yields
. Because answer
is
, that cannot be the answer, so we choose choice
.
Solution 5 (Process of Elimination)
We simply look at the units digit of the problem we have (or take mod )
Since the only answer with
in the units digit is
, We can then continue if you are desperate to use guess and check or a actually valid method to find the answer is
.
Solution 6 (Faster Distribution)
Observe that and
~laythe_enjoyer211
Solution 7 (Cubes)
Let . Then, we have
~erics118
Solution 8 (Super Fast)
It's not hard to observe and express into
, and
into
.
We then simplify the original expression into , which could then be simplified into
, which we can get the answer of
.
~RULE101
Video Solution (⚡️ 1 min solve ⚡️)
~Education, the Study of Everything
Video Solution by Pi Academy
https://youtu.be/GPoTfGAf8bc?si=JYDhLVzfHUbXa3DW
Video Solution by FrankTutor
https://www.youtube.com/watch?v=ez095SvW5xI
Video Solution Daily Dose of Math
~Thesmartgreekmathdude
Video Solution 1 by Power Solve
https://www.youtube.com/watch?v=j-37jvqzhrg
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=6SQ74nt3ynw