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National and Regional Contests
Puerto Rico Contests
Puerto Rico Team Selection Test
2021 Puerto Rico Team Selection Test
6
6
Part of
2021 Puerto Rico Team Selection Test
Problems
(1)
max no allied of 2021, product of primes Puerto Rico TST 2021.6
Source:
9/22/2021
Two positive integers
n
,
m
≥
2
n,m\ge 2
n
,
m
≥
2
are called allies if when written as a product of primes (not necessarily different):
n
=
p
1
p
2
.
.
.
p
s
n=p_1p_2...p_s
n
=
p
1
p
2
...
p
s
and
m
=
q
1
q
2
.
.
.
q
t
m=q_1q_2...q_t
m
=
q
1
q
2
...
q
t
, turns out that:
p
1
+
p
2
+
.
.
.
+
p
s
=
q
1
+
q
2
+
.
.
.
+
q
t
p_1 + p_2 + ... + p_s = q_1 + q_2 + ... + q_t
p
1
+
p
2
+
...
+
p
s
=
q
1
+
q
2
+
...
+
q
t
(a) Show that the biggest ally of any positive integer has to have only
2
2
2
and
3
3
3
in its prime factorization. (b) Find the biggest number which is allied of
2021
2021
2021
.
number theory
primes