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Taiwan National Olympiad
1997 Taiwan National Olympiad
6
6
Part of
1997 Taiwan National Olympiad
Problems
(1)
number of the form $2^p3^q$
Source: 6-th Taiwanese Mathematical Olympiad 1997
1/18/2007
Show that every number of the form
2
p
3
q
2^{p}3^{q}
2
p
3
q
, where
p
,
q
p,q
p
,
q
are nonnegative integers, divides some number of the form
a
2
k
1
0
2
k
+
a
2
k
−
2
1
0
2
k
−
2
+
.
.
.
+
a
2
1
0
2
+
a
0
a_{2k}10^{2k}+a_{2k-2}10^{2k-2}+...+a_{2}10^{2}+a_{0}
a
2
k
1
0
2
k
+
a
2
k
−
2
1
0
2
k
−
2
+
...
+
a
2
1
0
2
+
a
0
, where
a
2
i
∈
{
1
,
2
,
.
.
.
,
9
}
a_{2i}\in\{1,2,...,9\}
a
2
i
∈
{
1
,
2
,
...
,
9
}
number theory unsolved
number theory