Difference between revisions of "2017 UNCO Math Contest II Problems/Problem 11"
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== Problem == | == Problem == | ||
| + | Divide and Conquer | ||
| + | (a) How many different factorizations are there of <math>4096</math> (which is <math>2^{12}</math>) | ||
| + | in which each factor is either a square or a cube (or both) of an integer and | ||
| + | each factor is greater than one? | ||
| + | Regard <math>4 \times 4 \times 4 \times 8 \times 8</math> and <math>4 \times 8 \times 4 \times 8 \times 4</math> as the same factorization: the order in which the factors are written does not matter. | ||
| + | Regard the number itself, <math>4096</math>, as one of the factorizations. | ||
| + | |||
| + | (b) How many different factorizations are there of <math>46,656</math> as a product of factors in which each | ||
| + | factor is either a square or a cube (or both) of an integer and each factor is greater than one? As | ||
| + | before, the order in which the factors is written does not matter, and the number itself counts | ||
| + | as a factorization. Note that <math>46,656</math> = <math>2^6 \times 3^6</math>. | ||
== Solution == | == Solution == | ||
Revision as of 00:18, 20 May 2017
Problem
Divide and Conquer
(a) How many different factorizations are there of 
(which is 
)
in which each factor is either a square or a cube (or both) of an integer and
each factor is greater than one?
Regard 
and 
as the same factorization: the order in which the factors are written does not matter.
Regard the number itself, , as one of the factorizations.
(b) How many different factorizations are there of 
as a product of factors in which each
factor is either a square or a cube (or both) of an integer and each factor is greater than one? As
before, the order in which the factors is written does not matter, and the number itself counts
as a factorization. Note that = 
.
Solution
See also
| 2017 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by last question | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
