Difference between revisions of "2010 AMC 12B Problems/Problem 9"
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== Solution == | == Solution == | ||
− | We know that <math>n^2 = k^3</math> and <math> n^3 = m^2 </math>. Cubing and squaring the equalities respectively gives <math> n^6 = k^9 = m^4 </math>. Let <math>a = n^6</math>. <math>a</math> must be a perfect <math>36</math>-th power because <math>lcm(9,4) = 36</math>, which means that <math>n</math> must be a perfect | + | We know that <math>n^2 = k^3</math> and <math> n^3 = m^2 </math>. Cubing and squaring the equalities respectively gives <math> n^6 = k^9 = m^4 </math>. Let <math>a = n^6</math>. Now we know <math>a</math> must be a perfect <math>36</math>-th power because <math>lcm(9,4) = 36</math>, which means that <math>n</math> must be a perfect <math>6</math>-th power. The smallest number whose sixth power is a multiple of <math>20</math> is <math>10</math>, because the only prime factors of <math>20</math> are <math>2</math> and <math>5</math>, and <math>10 = 2 \times 5</math>. Therefore our is equal to number <math>10^6 = 1000000</math>, with <math>7</math> digits <math>\Rightarrow \boxed {E}</math>. |
== See also == | == See also == | ||
{{AMC12 box|year=2010|num-b=8|num-a=10|ab=B}} | {{AMC12 box|year=2010|num-b=8|num-a=10|ab=B}} |
Revision as of 17:30, 18 February 2012
Problem 9
Let be the smallest positive integer such that
is divisible by
,
is a perfect cube, and
is a perfect square. What is the number of digits of
?
Solution
We know that and
. Cubing and squaring the equalities respectively gives
. Let
. Now we know
must be a perfect
-th power because
, which means that
must be a perfect
-th power. The smallest number whose sixth power is a multiple of
is
, because the only prime factors of
are
and
, and
. Therefore our is equal to number
, with
digits
.
See also
2010 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |