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Saint Petersburg Mathematical Olympiad
2013 Saint Petersburg Mathematical Olympiad
7
Divisors of naturals
Divisors of naturals
Source: St Petersburg Olympiad 2013, Grade 10, P7
October 13, 2017
number theory
Problem Statement
Let
a
1
,
a
2
a_1,a_2
a
1
,
a
2
- two naturals, and
1
<
b
1
<
a
1
,
1
<
b
2
<
a
2
1<b_1<a_1,1<b_2<a_2
1
<
b
1
<
a
1
,
1
<
b
2
<
a
2
and
b
1
∣
a
1
,
b
2
∣
a
2
b_1|a_1,b_2|a_2
b
1
∣
a
1
,
b
2
∣
a
2
. Prove that
a
1
b
1
+
a
2
b
2
−
1
a_1b_1+a_2b_2-1
a
1
b
1
+
a
2
b
2
−
1
is not divided by
a
1
a
2
a_1a_2
a
1
a
2
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