Difference between revisions of "2013 UNCO Math Contest II Problems/Problem 2"
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== Problem == | == Problem == | ||
+ | A number <math>x</math> is equal to <math>7\cdot24\cdot48</math>. What is the smallest positive integer <math>y</math> such that the product <math>xy</math> is a perfect cube? | ||
− | + | == Solution == | |
+ | We can factor <math>12</math> into <math>2 \times 2 \times 3</math>. There are already two factors of two, so we only need to multiply it by <math>3</math> to get two factors of three, giving us <math>36</math>. | ||
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− | + | To find the perfect cube, we need all of the prime factors to be to the third power. Because <math>2</math> is squared, we need to multiply by a power of <math>2</math>, giving us <math>2 \times 12</math>, which is <math>24</math>. Because we only have one power of three, we need two more, so we multiply <math>24 \times 3 \times 3</math>, giving us <math>216</math>, which is a perfect cube. | |
− | + | To find a perfect 6th power, we multiply <math>36</math> by <math>216</math> to get <math>7776</math>. We know that the factorization of this number is <math>2^5 \times 3^5</math>. This means that this number is <math>6^5</math>. We need a perfect 6th, so we multiply by <math>6</math> to get <math>46656</math>, which is <math>6^6</math>, or <math>\boxed{588}</math>. | |
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== See Also == | == See Also == |
Latest revision as of 19:47, 11 December 2023
Problem
A number is equal to
. What is the smallest positive integer
such that the product
is a perfect cube?
Solution
We can factor into
. There are already two factors of two, so we only need to multiply it by
to get two factors of three, giving us
.
To find the perfect cube, we need all of the prime factors to be to the third power. Because is squared, we need to multiply by a power of
, giving us
, which is
. Because we only have one power of three, we need two more, so we multiply
, giving us
, which is a perfect cube.
To find a perfect 6th power, we multiply by
to get
. We know that the factorization of this number is
. This means that this number is
. We need a perfect 6th, so we multiply by
to get
, which is
, or
.
See Also
2013 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNCO Math Contest Problems and Solutions |