Difference between revisions of "2018 AMC 10B Problems/Problem 11"
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<math>\textbf{(A) } p^2+16 \qquad \textbf{(B) } p^2+24 \qquad \textbf{(C) } p^2+26 \qquad \textbf{(D) } p^2+46 \qquad \textbf{(E) } p^2+96</math> | <math>\textbf{(A) } p^2+16 \qquad \textbf{(B) } p^2+24 \qquad \textbf{(C) } p^2+26 \qquad \textbf{(D) } p^2+46 \qquad \textbf{(E) } p^2+96</math> | ||
| − | ==Solution 1== | + | ==Solution 1 (Mod)== |
Because squares of a non-multiple of 3 is always <math>1 \pmod{3}</math>, the only expression is always a multiple of <math>3</math> is <math>\boxed{\textbf{(C) } p^2+26} </math>. This is excluding when <math>p\equiv 0 \pmod{3}</math>, which only occurs when <math>p=3</math>, then <math>p^2+26=35</math> which is still composite. | Because squares of a non-multiple of 3 is always <math>1 \pmod{3}</math>, the only expression is always a multiple of <math>3</math> is <math>\boxed{\textbf{(C) } p^2+26} </math>. This is excluding when <math>p\equiv 0 \pmod{3}</math>, which only occurs when <math>p=3</math>, then <math>p^2+26=35</math> which is still composite. | ||
Revision as of 22:23, 20 October 2021
Contents
Problem
Which of the following expressions is never a prime number when 
is a prime number?
Solution 1 (Mod)
Because squares of a non-multiple of 3 is always 
, the only expression is always a multiple of 
is 
. This is excluding when 
, which only occurs when 
, then 
which is still composite.
Solution 2 (Answer Choices)
Since the question asks which of the following will never be a prime number when is a prime number, a way to find the answer is by trying to find a value for such that the statement above won't be true.
A) 
isn't true when 
because 
, which is prime
B) 
isn't true when 
because 
, which is prime
C) 
D) 
isn't true when because 
, which is prime
E) 
isn't true when 
because 
, which is prime
Therefore, 
is the correct answer.
-DAWAE
Minor edit by Lucky1256. P=___ was the wrong number.
More minor edits by beanlol.
Video Solution
~savannahsolver
Video Solution
https://youtu.be/3bRjcrkd5mQ?t=187
~ pi_is_3.14
See Also
| 2018 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 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. 
