Test of Mathematics Solution Subjective 35 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
Also see: Cheenta I.S.I. & C.M.I. Entrance Course
(a) Prove that, for any odd integer n, when divided by
always leaves remainder
.
(b) Hence or otherwise show that we cannot find integers such that
.
For part (a) we consider and expand it's fourth power binomially to get
Now ; if
is even then
is divisible by
and if
is odd
is even and product of
and
is divisible by
. Since
is already divisible by
we conclude
when divided by
gives
as remainder.
For part (b) we note that when divided by
, produces
as the remainder. Each of the eight of fourth powers when divided by
produces either
(when
is even) or
(when
is odd using part (a)) as remainder. Thus they can add up to at most
(modulo
) hence can never be equal to
(which
is modulo
).
Test of Mathematics Solution Subjective 35 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
Also see: Cheenta I.S.I. & C.M.I. Entrance Course
(a) Prove that, for any odd integer n, when divided by
always leaves remainder
.
(b) Hence or otherwise show that we cannot find integers such that
.
For part (a) we consider and expand it's fourth power binomially to get
Now ; if
is even then
is divisible by
and if
is odd
is even and product of
and
is divisible by
. Since
is already divisible by
we conclude
when divided by
gives
as remainder.
For part (b) we note that when divided by
, produces
as the remainder. Each of the eight of fourth powers when divided by
produces either
(when
is even) or
(when
is odd using part (a)) as remainder. Thus they can add up to at most
(modulo
) hence can never be equal to
(which
is modulo
).