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Israel National Olympiad
2024 Israel National Olympiad (Gillis)
P2
P2
Part of
2024 Israel National Olympiad (Gillis)
Problems
(1)
Sum of fractional parts
Source: 2024 Israel National Olympiad (Gillis) P2
12/7/2023
A positive integer
x
x
x
satisfies the following:
{
x
3
}
+
{
x
5
}
+
{
x
7
}
+
{
x
11
}
=
248
165
\{\frac{x}{3}\}+\{\frac{x}{5}\}+\{\frac{x}{7}\}+\{\frac{x}{11}\}=\frac{248}{165}
{
3
x
}
+
{
5
x
}
+
{
7
x
}
+
{
11
x
}
=
165
248
Find all possible values of
{
2
x
3
}
+
{
2
x
5
}
+
{
2
x
7
}
+
{
2
x
11
}
\{\frac{2x}{3}\}+\{\frac{2x}{5}\}+\{\frac{2x}{7}\}+\{\frac{2x}{11}\}
{
3
2
x
}
+
{
5
2
x
}
+
{
7
2
x
}
+
{
11
2
x
}
where
{
y
}
\{y\}
{
y
}
denotes the fractional part of
y
y
y
.
number theory
remainders
fractional part
national olympiad