Difference between revisions of "2000 PMWC Problems/Problem I4"
(Created page with "==Problem== Given that <math>A^4=75600\times B</math>. If <math>A</math> and <math>B</math> are positive integers, find the smallest value of <math>B</math>. ==Solution== ==See...") |
(→Solution) |
||
| Line 3: | Line 3: | ||
==Solution== | ==Solution== | ||
| + | Given that <math>A^4=75600\times B</math>. If <math>A</math> and <math>B</math> are positive integers, find the smallest value of <math>B</math>. | ||
| + | |||
| + | |||
| + | If <math>A</math> and <math>B</math> are positive integers, then <math>B</math> must be the smallest positive integer that, when multiplied by <math>75600</math>, yields a perfect fourth power. The prime factorization of <math>75600</math> is <math>2^4 * 3^3 * 5^2 * 7^1</math>, so the smallest value of <math>B</math> is <math>2^0 * 3^1 * 5^2 * 7^3 = 25,725</math>. | ||
==See Also== | ==See Also== | ||
Revision as of 10:58, 23 December 2019
Problem
Given that 
. If 
and 
are positive integers, find the smallest value of .
Solution
Given that . If and are positive integers, find the smallest value of .
If and are positive integers, then must be the smallest positive integer that, when multiplied by 
, yields a perfect fourth power. The prime factorization of is 
, so the smallest value of is 
.
