The problem involves finding the ratio of two natural numbers \(a\) and \(b\) given the equation \(a \times b = 1364 + (a + b)\) with the conditions that \(a > b\) and either \(a\) or \(b\) is a perfect square. The solution reveals that \(a = 52\) and \(b = 26\), leading to the ratio \(a:b = 2:1\). The perfect square in this case is \(b\), which equals \(26\) (since \(5^2 = 25\) is the closest perfect square less than \(26\)).
PREREQUISITESMathematics students, educators, and anyone interested in solving algebraic equations involving natural numbers and perfect squares.
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 = $49::2$