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? | |
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== Solution == | == Solution == | ||
| − | We can factor 12 into 2 | + | 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>. |
| − | To find the perfect cube, we need all of the prime factors to be to the third power. Because 2 is squared, we need to multiply by a power of 2, giving us 2 | + | 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 36 | + | 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>. |
== See Also == | == See Also == | ||
Latest revision as of 18:47, 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 | ||
