Directions : Solve the number series to answer the following questions. ⁿ√A 29…
2023
Directions : Solve the number series to answer the following questions.
ⁿ√A 29 B C 437 B + 687 1294 / ⁿ√A
Note:
(i) ‘n’ is a positive integer.
(ii) Difference between 29 and B is X, which has 3 factors excluding X itself, those are 13, 5 and A but when we include X then number of factors become 4.
What is the LCM of B + 20 and A?
- A.
101
- B.
102
- C.
114
- D.
104
- E.
115
Show answer & explanation
Correct answer: C
Concept
A natural number has exactly 4 divisors only when it is either a product of two distinct primes (p × q) or the cube of a single prime (p3). Listing all divisors and then removing the number itself leaves exactly 3 smaller divisors. Also, the LCM of any number with 1 is the number itself, because 1 divides every integer.
Application
Let X be the difference between 29 and B. The clue says X has exactly 4 divisors in total, and removing X itself leaves the three divisors 13, 5 and A.
Having exactly 4 divisors with two of them being the distinct primes 5 and 13 forces the form X = p × q, whose full divisor list is 1, p, q and pq. So the three divisors below X are 1, 5 and 13; matching this with the given 13, 5 and A gives A = 1.
Hence X = 5 × 13 = 65, and its four divisors are 1, 5, 13 and 65.
X = 65 is the gap between 29 and B, so B is either 29 + 65 = 94 or 29 − 65 = −36. Every term in this series is a positive quantity and the terms increase along the row (29, then B, then C, then 437), so the negative option is rejected, leaving B = 94; this also makes B + 687 = 781 fall correctly between 437 and 1294.
Now B + 20 = 94 + 20 = 114 and A = 1.
LCM(114, 1) = 114, because 1 divides 114.
Cross-check
The divisors of 65 are 1, 5, 13 and 65 - exactly four - and dropping 65 leaves 1, 5, 13, which matches A = 1. With A = 1 the n-th root term equals 1 for every positive integer n, keeping the series self-consistent. Therefore LCM(B + 20, A) = 114.