Difference between revisions of "1985 AIME Problems/Problem 7"
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Assume that <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are [[positive integer]]s such that <math>a^5 = b^4</math>, <math>c^3 = d^2</math>, and <math>c - a = 19</math>. Determine <math>d - b</math>. | Assume that <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are [[positive integer]]s such that <math>a^5 = b^4</math>, <math>c^3 = d^2</math>, and <math>c - a = 19</math>. Determine <math>d - b</math>. | ||
== Solution == | == Solution == | ||
− | It follows from the givens that <math>a</math> is a [[perfect power | perfect fourth power]], <math>b</math> is a perfect fifth power, <math>c</math> is a [[perfect square]] and <math>d</math> is a [[perfect cube]]. Thus, there exist [[integer]]s <math>s</math> and <math>t</math> such that <math>a = t^4</math>, <math>b = t^5</math>, <math>c = s^2</math> and <math>d = s^3</math>. So <math>s^2 - t^4 = 19</math>. We can factor the left-hand side of this [[equation]] as a difference of two squares, <math>(s - t^2)(s + t^2) = 19</math>. 19 is a [[prime number]] and <math>s + t^2 > s - t^2</math> so we must have <math>s + t^2 = 19</math> and <math>s - t^2 = 1</math>. Then <math>s = 10, t = 3</math> and so <math>d = s^3 = 1000</math>, <math>b = t^5 = 243</math> and <math>d - b = | + | It follows from the givens that <math>a</math> is a [[perfect power | perfect fourth power]], <math>b</math> is a perfect fifth power, <math>c</math> is a [[perfect square]] and <math>d</math> is a [[perfect cube]]. Thus, there exist [[integer]]s <math>s</math> and <math>t</math> such that <math>a = t^4</math>, <math>b = t^5</math>, <math>c = s^2</math> and <math>d = s^3</math>. So <math>s^2 - t^4 = 19</math>. We can factor the left-hand side of this [[equation]] as a difference of two squares, <math>(s - t^2)(s + t^2) = 19</math>. 19 is a [[prime number]] and <math>s + t^2 > s - t^2</math> so we must have <math>s + t^2 = 19</math> and <math>s - t^2 = 1</math>. Then <math>s = 10, t = 3</math> and so <math>d = s^3 = 1000</math>, <math>b = t^5 = 243</math> and <math>d-b=\boxed{757}</math>. |
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+ | == Video Solution by OmegaLearn == | ||
+ | https://youtu.be/euz1azVKUYs?t=709 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
== See also == | == See also == |
Latest revision as of 11:04, 4 November 2022
Problem
Assume that ,
,
, and
are positive integers such that
,
, and
. Determine
.
Solution
It follows from the givens that is a perfect fourth power,
is a perfect fifth power,
is a perfect square and
is a perfect cube. Thus, there exist integers
and
such that
,
,
and
. So
. We can factor the left-hand side of this equation as a difference of two squares,
. 19 is a prime number and
so we must have
and
. Then
and so
,
and
.
Video Solution by OmegaLearn
https://youtu.be/euz1azVKUYs?t=709
~ pi_is_3.14
See also
1985 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |