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Joint Entrance Examination
Mathematics
Binomial Theorem
Question 179
Question 179 of 347
Medium
NUMERICAL
If
(
2021
)
3762
(2021)^{3762}
(
2021
)
3762
is divided by 17, then the remainder is Answer: 4
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Solutions · 1
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Official solution
2021
=
17
×
119
−
2
2021=17\times119-2
2021
=
17
×
119
−
2
, so
2021
≡
−
2
(
m
o
d
17
)
2021\equiv-2\pmod{17}
2021
≡
−
2
(
mod
17
)
:
(
2021
)
3762
≡
(
−
2
)
3762
=
2
3762
(
m
o
d
17
)
(2021)^{3762}\equiv(-2)^{3762}=2^{3762}\pmod{17}
(
2021
)
3762
≡
(
−
2
)
3762
=
2
3762
(
mod
17
)
Since
2
4
=
16
≡
−
1
(
m
o
d
17
)
2^4=16\equiv-1\pmod{17}
2
4
=
16
≡
−
1
(
mod
17
)
,
2
8
≡
1
(
m
o
d
17
)
2^8\equiv1\pmod{17}
2
8
≡
1
(
mod
17
)
.
3762
=
8
×
470
+
2
3762=8\times470+2
3762
=
8
×
470
+
2
, so
2
3762
≡
2
2
=
4
(
m
o
d
17
)
2^{3762}\equiv2^2=4\pmod{17}
2
3762
≡
2
2
=
4
(
mod
17
)
. Remainder = 4
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