Given below are two number series I and II where the missing numbers in series 'I' and 'II' are 'P' and 'Q', respectively. Find the value of 'P' and 'Q' and find which among the gi... Given below are two number series I and II where the missing numbers in series 'I' and 'II' are 'P' and 'Q', respectively. Find the value of 'P' and 'Q' and find which among the given options gives the L.C.M of 'P' and 'Q'. I: 77, 106, 48, P, 19, 164 II: 141, 98, Q, 63, 71, 96 Read more

Steps to Solve

1. Identify Patterns in Series I

   The first series is: $77, 106, 48, P, 19, 164$. Let's find a pattern or relationship in the existing numbers. Looking at the differences:

   • $106 - 77 = 29$
   • $48 - 106 = -58$
   • To find $P$, we can check the transition to the next number (19). Noticing that $19 - P$ should relate to previous terms:
   • The differences seem to vary a lot. Another approach could be identifying if the series has alternating patterns.

2. Calculate the Missing Value P

   Assume a relationship involves subtracting a number:

   • $77 - 106 = -29$
   • $106 - 48 = 58$
   • If we consider $P$ as $x$, and find that $19 < P$, we can estimate or assume $P$ in relation to $19$. It's difficult to see a direct clear path; let’s note down the numbers and work with the second series to find the missing Q.

3. Identify Patterns in Series II

   The second series is: $141, 98, Q, 63, 71, 96$. Again, look for differences:

   • $98 - 141 = -43$
   • $63 - 98 = -35$
   • Assume similar transitions might exist as we previously did. We'll need to express how $Q$ interacts between those values.

4. Calculate the Missing Value Q

   Let $Q = y$. Then we have:

   • The difference pattern indicates possible values, so:
   • Estimating, based on relationships, it makes sense $Q$ should align lower than previous twice, around $80$.

5. Finding L.C.M. of P and Q

   Let's finalize with potential $P = 78$ and $Q = 78$ or other values. Calculate their L.C.M.:

   • The least common multiple $L = \text{lcm}(P, Q)$ can be deduced. For examples, if $P = 78$ and $Q = 63$, similar methods yield optimal solutions based on their factors.

6. Choose from Given Options

   Comparing potential results against options [1] 1080, [2] 1200, [3] 724, [4] 960, and [5] 1440, we can find correct matches based on decomposed factors of determined L.C.M.