Difference between revisions of "2019 AIME I Problems/Problem 14"
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<math>\phi(p)</math> is called the "Euler Function" of integer <math>p</math>. | <math>\phi(p)</math> is called the "Euler Function" of integer <math>p</math>. | ||
− | Euler theorem: define <math>\ | + | Euler theorem: define <math>\phili(p)</math> as the number of positive integers less than <math>n</math> but relatively prime to <math>n</math>, then we have <cmath>\phi(p)=p\cdot \prod^n_{i=1}(1-\frac{1}{p_i})</cmath> where <math>p_1,p_2,...,p_n</math> are the prime factors of <math>p</math>. Then, we have <cmath>a^{\phi(p)} \equiv 1\ (\mathrm{mod}\ p)</cmath> if <math>(a,p)=1</math>. |
Furthermore, <math>ord_n(a)</math> for an integer <math>a</math> relatively prime to <math>n</math> is defined as the smallest positive integer <math>d</math> such that <math>a^{d} \equiv 1\ (\mathrm{mod}\ n)</math>. An important property of the order is that <math>ord_n(a)|\phi(n)</math>. | Furthermore, <math>ord_n(a)</math> for an integer <math>a</math> relatively prime to <math>n</math> is defined as the smallest positive integer <math>d</math> such that <math>a^{d} \equiv 1\ (\mathrm{mod}\ n)</math>. An important property of the order is that <math>ord_n(a)|\phi(n)</math>. |
Revision as of 20:41, 25 September 2019
Problem 14
Find the least odd prime factor of .
Solution 1
The problem tells us that for some prime
. We want to find the smallest odd possible value of
. By squaring both sides of the congruence, we get
.
Since ,
=
or
However, if =
or
then
clearly will be
instead of
, causing a contradiction.
Therefore, . Because
,
is a multiple of 16. Since we know
is prime,
or
. Therefore,
must be
. The two smallest primes that are
are
and
.
, but
, so our answer is
.
Note to solution 1
is called the "Euler Function" of integer
.
Euler theorem: define $\phili(p)$ (Error compiling LaTeX. Unknown error_msg) as the number of positive integers less than
but relatively prime to
, then we have
where
are the prime factors of
. Then, we have
if
.
Furthermore, for an integer
relatively prime to
is defined as the smallest positive integer
such that
. An important property of the order is that
.
Video Solution
On The Spot STEM:
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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