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2021 ISI Entrance Examination
3
3
Part of
2021 ISI Entrance Examination
Problems
(1)
Unique Rational Number Representation
Source: ISI 2021 P3
7/18/2021
Prove that every positive rational number can be expressed uniquely as a finite sum of the form
a
1
+
a
2
2
!
+
a
3
3
!
+
⋯
+
a
n
n
!
,
a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},
a
1
+
2
!
a
2
+
3
!
a
3
+
⋯
+
n
!
a
n
,
where
a
n
a_n
a
n
are integers such that
0
≤
a
n
≤
n
−
1
0 \leq a_n \leq n-1
0
≤
a
n
≤
n
−
1
for all
n
>
1
n > 1
n
>
1
.
number theory
algebra
power series
isi