MHB Least Possible Value of a+b: 11 & 13 Divisibility

[kaliprasad]

find the least possible value of a + b where a and b are positive integers and 11 divides $a+ 13b$
and 13 divides $a + 11 b$

 

[Albert1]

find the least possible value of a + b where a and b are positive and 11 divides $a+ 13b$
and 13 divides $a + 11 b$

Spoiler

the least possible value of a + b=14
a=11.5,b=2.5

 

[kaliprasad]

Spoiler

the least possible value of a + b=14
a=11.5,b=2.5

sorry I meant integers

 

[Albert1]

sorry I meant integers

Spoiler

if $a,b\in N$
$min(a+b)=28, a=23,b=5$

 

[kaliprasad]

Albert has provided the answer
here is the solution

Spoiler

$11$ divides $a + 2b$ and hence $11$ divides $6a + 12b$ or $11$ divides $6a + b$
$13$ divides $a - 2b$ and hence $13$ divides $6a - 12b$ or $13$ divides $6a + b$

so $6a + b$ is divisible by $11$ and $13$ and hence $143$
say $6a +b = 143 t\cdots(1)$
$6a + 6b = 143t + 5b = 144 t + 6b - ((t+b)$
So $t + b$ is divisible by $6$ and hence $t + b > 6 \cdots(2)$
$(a+b) = 143t + 5b = 138 t + 5(t+b) >=168$
Hence $a + b >= 28$
From (2) putting $t = 1$ we get $b= 5$ and from (1) we get $a = 23$ so $a=23$ and $b=5$ satisfies
the condition so $a+b$ lowest value is 28