Question

Find the largest positive integer that will divide $150,187$ and leaving remainders $6,7$ and $11$ respectively.

Answer 1

It is given that on dividing $150$ by the required number, the remainder will be $6$.
$⟹150-6=144$, will be exactly divisible by the required largest positive integer.
Hence, the required number is a factor of $144$.

Similarly, required positive integer is a factor of
$187-7=180$ and $203-11=192$.

Hence, the required positive integer is the HCF of $144,180$ and $192$

.
By prime factorization:

$144=2^4\times 3^2$

$180=2^2\times 3^2\times 5$

$192=2^6\times 3$

$\therefore \text{HCF}(144,180,192)=2^2\times 3=12$

Hence, the required positive integer is $12$.
$\therefore$ $12$ is the largest positive integer that will divide $150,187$ and $203$ leaving remainders $6,7$ and $11$ respectively.