Difference between revisions of "2021 CIME I Problems/Problem 2"
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For digits <math>a, b, c,</math> with <math>a\neq 0,</math> the positive integer <math>N</math> can be written as <math>\underline{a}\underline{a}\underline{b}\underline{b}</math> in base <math>9,</math> and <math>\underline{a}\underline{a}\underline{b}\underline{b}\underline{c}</math> in base <math>5</math>. Find the base-<math>10</math> representation of <math>N</math>. | For digits <math>a, b, c,</math> with <math>a\neq 0,</math> the positive integer <math>N</math> can be written as <math>\underline{a}\underline{a}\underline{b}\underline{b}</math> in base <math>9,</math> and <math>\underline{a}\underline{a}\underline{b}\underline{b}\underline{c}</math> in base <math>5</math>. Find the base-<math>10</math> representation of <math>N</math>. | ||
Latest revision as of 19:00, 24 July 2024
Problem
For digits with
the positive integer
can be written as
in base
and
in base
. Find the base-
representation of
.
Solution
Consider the different representations of the number and equate them:
Note that c can't contribute since it is less than 5 so
Next note that
since
and
the only solution is
,
Thus in base 10 the number is