what natural numbers are less than or equal to 7 that are relatively prime?
Trouble
Permit be an integer and
be all the natural numbers less than
and relatively prime number to
. If
prove that
must be either a prime or a ability of
.
Solution 1
We utilize Bertrand's Postulate: for is a positive integer, there is a prime number in the interval
.
Clearly, must be equal to the smallest prime number
which does not divide
. If
, then
is a prime since the common deviation
is equal to
, i.east. all positive integers less than
are coprime to
. If, on the ther hand,
, we find
to exist a power of
: the positive integers less than
and coprime to it are precisely the odd ones. We may thus assume that
. Furthermore, since
, the positive integers less than
and coprime to it cannot be
solitary, i.e.
.
Past Bertrand'southward Postulate, the largest prime less than is strictly larger than
, and so it cannot divide
. We will denote this prime by
. We know that
. It'south easy to check that for
there is a prime
strictly between
.
, then, in detail,
. On the other hand, if
, the two largest primes
which are smaller than
satisfy
(once more, Bertrand's Postulate), so
so, one time over again, nosotros have
(this is what I wanted to evidence here).
Now, , and all the numbers
are smaller than
(they are smaller than
, which is smaller than
, co-ordinate to the paragraph above). I of these numbers, however, must exist divisible by
. Since our hypothesis tells u.s.a. that all of them must be coprime to
, nosotros have a contradiction.
This solution was posted and copyrighted by grobber. The original thread for this trouble can be found here: [1]
Solution 2
Let . So nosotros have
. Since
, and then
is the largest positive integer smaller than
and co-prime to
. Thus,
and
which shows
and so
divides
. We make ii cases. Case i:
is odd. Then
. So, every divisor of
is co-prime to
also, and so is
. Thus, there is a
such that
which implies
. Therefore, in this case,
and we easily notice that all numbers less than
are co-prime to
, so
must be a prime number. Case 2:
is even. But and then
and also
is ruled out. And so
. Say,
. Then,
divides
. If
is odd,
and
forcing
, contradiction! And so,
with
. If
for some
, then
. And so, once more
for a
. If
is odd, then
and thus,
. Similarly, we discover that actually
must occur. Nosotros finally have,
. Just then all odd numbers less than
are co-prime to
. Then,
does not take any odd factor i.due east.
for some
.
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Solution 3
Nosotros use Bonse's Inequality: , for all
,
.
Let denote the smallest prime which does not divide
.
Example 1) . This implies
is a prime. Case 2)
. This implies
for some
. Case iii)
. Since
, we have that
. Therefore
, but
. Case 4)
. Write
. We have
. Hence
, a contradiction since
.
This solution was posted and copyrighted by SFScoreLow. The original thread for this trouble can be constitute hither: [3]
Solution 4
Clearly, . Now permit
be the smallest prime number which is non a divisor of
. Hence,
, and the mutual divergence
. Find that since
, we take that
, or
. This ways that all prime number factors of
are prime factors of
. Notwithstanding, due to the minimality of
, all primes factors of
are prime factors of
. By the Euclidean algorithm, the only prime factors of
are
, and then
or
, by the theory of Fermat primes. We will now do casework on
.
Instance 1: This case is relatively easy.
, then all numbers less than
must be relatively prime to
. Nevertheless, if
, where
and
, observe that
is not relatively prime to
, so
must be prime for this example to work.
Case ii: and
. This case is somewhat tricky. Notice that
is even. Since
, we take that all numbers less than
which are odd are relatively prime number to
. Still, if
has a odd divisor
, clearly
is not relatively prime to
, then
must exist a power of two.
Case 3: . Note that
must be a divisor of
, so
cannot exist a multiple of
for any
. I will contradict this argument. First, I merits that
for all integers
.
is piddling, and then let me show that
has just solutions less than or equal to
. Since
is a multiplicative function, observe that
cannot exist a multiple of a prime greater than
, considering for such primes
we have that
, which is contradiction. Hence,
is just composed of factors of
and
. If information technology divides both, and is of the form
for
and
positive, nosotros get that nosotros demand
, and so
and
in this case. If it is of the form
, nosotros get
. And if it is of the course
, we get
.
,
, and
are all less than or equal to
, so
. Now we're nearly washed. Since
, the number
must be in this sequence. However,
is of the grade
, which is a multiple of
, and we have contradiction, so we are washed.
This solution was posted and copyrighted by va2010. The original thread for this problem tin can be constitute here: [iv]
Solution 5
Clearly that . Allow
then
We can see that if
is a prime and then
satisfies the status. Nosotros consider the case when
is a composite number. Permit
be the smallest prime number divisor of
then
. Note that
for all prime number
since
for all
. Therefore, if
so there will exists
such that
or
, a contradiction since
. Thus,
. We consider two cases:
If and then
. Hence,
. This means there exists a prime
such that
. Since
then nosotros get
or
, a contradiction since
. Thus,
or
.
If then all odd numbers
are relatively prime number to
. This can only happen when
.
If and so
since
. Thus,
or there exist a prime number
such that
. Note that
and then
, a contradiction.
Thus, must be either a prime number or a ability of
.
This solution was posted and copyrighted by shinichiman. The original thread for this problem tin be found here: [5]
Encounter Likewise
Source: https://artofproblemsolving.com/wiki/index.php/1991_IMO_Problems/Problem_2
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