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