The discussion revolves around finding the ratio of two natural numbers \(a\) and \(b\) under specific conditions, including their product equating to a sum involving 1364, and the stipulation that either \(a\) or \(b\) is a perfect square. The scope includes mathematical reasoning and problem-solving related to number theory.
There is no consensus on which number, \(a\) or \(b\), is the perfect square, and the discussion remains unresolved regarding the specific values of \(a\) and \(b\) and their ratio.
The discussion does not clarify the assumptions regarding the nature of the perfect square or the methods to derive the ratio \(a:b\). There may be missing mathematical steps or definitions that are not fully explored.
Albert said:if :
(1)$a,b\in N $, and $a>b$
(2)$a\times b =1364+(a+b)$
(3) eather $a$ or $b$ is a perfect square
can you tell which one is a perfect square ? ( $a$ or $b$ ?),and please find $a:b$
kaliprasad said:$ab-a-b = 1364$
or $ab-a-b+1 = 1365$
or $(a-1)(b-1)= 1365= 3 *5 * 7 * 13$
now we get 3 choices
choice 1
$ a= 92, b= 16$ and ratio $a::b = 23::4$
choice 2
$ a= 40, b = 36$ and $a::b = 10::9 $
choice 3
$a = 256, b= 8$ and $a::b= 32::1$
Albert said:we get 4 choices
an you got two points
10:9 and 23:4
Albert said:we get 4 choices
an you got two points
10:9 and 23:4
correction:kaliprasad said:I got 3 choices and missed the 4th
$b= 4,a = 356$ giving $a::b= 89::1$
it should beAlbert said:correction:
$a:b=32:1$ this answer is not correct
$a:b=89:1$ this answer is not correct
$a:b=23:4$ this answer is correct
$a:b=10:9$ this answer is correct
bingo !kaliprasad said:it should be
$a=456,b=4$ ratio = $114::1$
$a=196,b=8$ ratio = $49::2$