Difference between revisions of "Euclid 2019/Problem 5"
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| + | (a) <math>\sqrt(50)=5\sqrt(2)</math>, and <math>5\sqrt(2)</math> can be expressed as <math>\sqrt(2)+4\sqrt(2)</math>, or <math>2\sqrt(2)+3\sqrt(2)</math>. | ||
| + | From the first equation, we get that <math>\sqrt(2)+\sqrt(32)</math>, and from the second of these equations, we have <math>\sqrt(8)+\sqrt(18)</math>. So our solutions for <math>(a,b)</math> are <math>\boxed{(2,32), (8,18)}</math>. | ||
Latest revision as of 14:25, 13 October 2025
Problem
(a) Determine the two pairs of positive integers 
with 
that satisfy the equation 
.
(b) Consider the system of equations: 
and 
. Determine the number of integers 
with 
for which there is at least one pair of integers 
that is a solution to the system.
Solution
(a) 
, and 
can be expressed as 
, or 
.
From the first equation, we get that 
, and from the second of these equations, we have 
. So our solutions for are 
.
