Difference between revisions of "2014 AIME I Problems/Problem 8"
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Since <math>N</math> and <math>N-1</math> are coprime, the only solutions are when <math>(N\mod{16},N\mod{625})\in\{(0,0),(0,1),(1,0),(1,1)\}</math>. | Since <math>N</math> and <math>N-1</math> are coprime, the only solutions are when <math>(N\mod{16},N\mod{625})\in\{(0,0),(0,1),(1,0),(1,1)\}</math>. | ||
− | Let <math>\varphi:\mathbb Z/10000\mathbb Z\to\mathbb Z/16\mathbb Z\times\mathbb Z/625\mathbb Z</math>, <math>x\mapsto (x\mod{16},x\mod{625})</math>. The statement of the Chinese Remainder theorem is that <math>\varphi</math> is an isomorphism between the two rings. In this language, the solutions are <math>\ | + | Let <math>\varphi:\mathbb Z/10000\mathbb Z\to\mathbb Z/16\mathbb Z\times\mathbb Z/625\mathbb Z</math>, <math>x\mapsto (x\mod{16},x\mod{625})</math>. The statement of the Chinese Remainder theorem is that <math>\varphi</math> is an isomorphism between the two rings. In this language, the solutions are <math>\varphi^{-1}(0,0)</math>, <math>\varphi^{-1}(0,1)</math>, <math>\varphi^{-1}(1,0)</math>, and <math>\varphi^{-1}(1,1)</math>. Now we easily see that <math>\varphi^{-1}(0,0)=0</math> and <math>\varphi^{-1}(1,1)=1</math>. Noting that <math>625\equiv 1\pmod{16}</math>, it follows that <math>\varphi^{-1}(1,0)=625</math>. To compute <math>\varphi^{-1}(0,1)</math>, note that <math>(0,1)=15(1,0)+(1,1)</math> in <math>\mathbb \mathbb Z/16\mathbb Z\times\mathbb Z/625\mathbb Z</math>, so since <math>\varphi</math> is linear in its arguments (by virtue of being an isomorphism), <math>\varphi^{-1}(0,1)=15\varphi^{-1}(1,0)+\varphi^{-1}(1,1)=15\times 625+1=9376</math>. |
The four candidate digit strings <math>abcd</math> are then <math>0000,0001,0625,9376</math>. Of those, only <math>9376</math> has nonzero first digit, and therefore the answer is <math>\boxed{937}</math>. | The four candidate digit strings <math>abcd</math> are then <math>0000,0001,0625,9376</math>. Of those, only <math>9376</math> has nonzero first digit, and therefore the answer is <math>\boxed{937}</math>. |
Revision as of 18:27, 14 March 2014
Problem 8
The positive integers and
both end in the same sequence of four digits
when written in base 10, where digit a is not zero. Find the three-digit number
.
Solution (bashing)
let for positive integer values t,a,b,c,d
when we square N we get that
However we dont have to deal with this whole expression but only with its last 4 digits so it is suffices to consider only:
know we need to compare each decimal digit with
and see whether the digits are congrount in base 10.
we first consider the ones digits:
this can happen for only 3 values : 1, 5 and 6
we can try to solve each case
- Case 1
considering the tenths place we have that:
so
considering the hundreds place we have that
so again
now considering the thousands place we have that
so we get
but
cannot be equal to 0 so we consider
- Case 2
considering the tenths place we have that:
( the extra 20 is carried from
which is equal to 25)
so
considering the hundreds place we have that
( the extra 100c is carried from the tenths place)
so
now considering the thousands place we have that
( the extra 1000b is carried from the hundreds place)
so a is equal 0 again
- Case 3
considering the tenths place we have that:
( the extra 20 is carried from
which is equal to 25)
if
then we have
so
considering the hundreds place we have that
( the extra 100c+100 is carried from the tenths place)
if then we have
so
now considering the thousands place we have that
( the extra 1000b+6000 is carried from the hundreds place)
if then we have
so
so we have that the last 4 digits of N are
and
is equal to
Solution (not bashing)
By the Chinese Remainder Theorem, the equation is equivalent to the two equations:
Since
and
are coprime, the only solutions are when
.
Let ,
. The statement of the Chinese Remainder theorem is that
is an isomorphism between the two rings. In this language, the solutions are
,
,
, and
. Now we easily see that
and
. Noting that
, it follows that
. To compute
, note that
in
, so since
is linear in its arguments (by virtue of being an isomorphism),
.
The four candidate digit strings are then
. Of those, only
has nonzero first digit, and therefore the answer is
.