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# If p is a prime number and q is a positive integer such that $p + q = 1696$. If p and q are co prime numbers and their LCM is 21879, then find p and q.

__Given__:

- $p + q = 1696$ ...(i)
- L.C.M of p and q $= 21879$
- p and q are co prime numbers and H.C.F of two co prime numbers is always 1.

__To find__:

We have to find the value of p and q.

__Solution__:

To find the values of p and q we need to use the below property:

- The product of two numbers is equal to the product of the highest common factor (H.C.F.) and lowest common multiple (L.C.M.) of those two numbers.

Using this property to get:

$p\ \times \ q\ =\ H.C.F\ \times \ L.C.M$

$p\ \times \ q\ =\ 1\ \times \ 21879$

$p\ \times \ q\ =\ 21879$

$q\ =\ \frac{21879}{p} \ \ \ \ \ \ \ \ ...(ii)$

Substituting value of q from equation (ii) into equation (I):

$p\ +\ \frac{21879}{p} \ =\ 1696$

$\frac{p^{2} \ +\ 21879}{p} \ =\ 1696$

$p^{2} \ +\ 21879\ =\ 1696p$

$p^{2} \ -\ 1696p\ +\ 21879\ =\ 0$

$p^{2} \ -\ 1683p\ -\ 13p\ +\ 21879\ =\ 0$

$p(p\ -\ 1683)\ -\ 13(p\ -\ 1683)\ =\ 0$

$(p\ -\ 1683)(p\ -\ 13)\ =\ 0$

$\mathbf{p\ =\ 1683,\ 13}$

It is already given in the problem that p is a prime number. So,

$p\ =\ 13$

Now, putting this value of p in equation (i) to get the value of q:

$13\ +\ q\ =\ 1696$

$q\ =\ 1696\ -\ 13$

$\mathbf{q\ =\ 1683}$

Therefore,

$\mathbf{p\ =\ 13\ \ and\ \ q\ =\ 1683}$.

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