Difference between revisions of "1990 AIME Problems/Problem 5"
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== Problem == | == Problem == | ||
| − | Let <math>n^{}_{}</math> be the smallest positive [[integer]] that is a multiple of <math>75_{}^{}</math> and has exactly <math>75_{}^{}</math> positive integral divisors, including <math>1_{}^{}</math> and itself. Find <math>n | + | Let <math>n^{}_{}</math> be the smallest positive [[integer]] that is a multiple of <math>75_{}^{}</math> and has exactly <math>75_{}^{}</math> positive integral divisors, including <math>1_{}^{}</math> and itself. Find <math>\frac{n}{75}</math>. |
== Solution == | == Solution == | ||
Revision as of 19:05, 29 January 2009
Problem
Let 
be the smallest positive integer that is a multiple of 
and has exactly positive integral divisors, including 
and itself. Find 
.
Solution
The prime factorization of 
. For 
to have exactly 
integral divisors, we need to have 
such that 
. Since 
, two of the prime factors must be 
and 
. To minimize , we can introduce a third prime factor, 
. Also to minimize , we want , the greatest of all the factors, to be raised to the least power. Therefore, 
and 
.
See also
| 1990 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
