MHB Find Ratio of a:b When $1364+(a+b)=a\times b$

[Albert1]

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]

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 $a:b$

Spoiler

$ab-a-b = 1364$
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 $a::b= 32::1$

 

[Albert1]

Spoiler

$ab-a-b = 1364$
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 $a::b= 32::1$

Spoiler

we get 4 choices
an you got two points
10:9 and 23:4

 

[MarkFL]

Spoiler

we get 4 choices
an you got two points
10:9 and 23:4

Albert,

Please use the spoiler tags when posting anything pertaining to a solution. Thanks! :)

 

[kaliprasad]

Spoiler

we get 4 choices
an you got two points
10:9 and 23:4

Spoiler

I got 3 choices and missed the 4th
$b= 4,a = 356$ giving $a::b= 89::1$

 

[Albert1]

Spoiler

I got 3 choices and missed the 4th
$b= 4,a = 356$ giving $a::b= 89::1$

correction:

Spoiler

$a:b=32:1$ 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

 

[kaliprasad]

correction:

Spoiler

$a:b=32:1$ 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

it should be

Spoiler

$a=456,b=4$ ratio = $114::1$
$a=196,b=8$ ratio = $49::2$

 

[Albert1]

it should be

Spoiler

$a=456,b=4$ ratio = $114::1$
$a=196,b=8$ ratio = $49::2$

bingo !