Shortcut Trick
The LCM of the first n natural numbers is the product of the highest prime powers ≤ n.
When extending the LCM from 110 to 115, we only look for new prime powers in the range 111 to 115.
The new numbers added are 111, 112, 113, 114, and 115.
113 is the only new prime number, and there are no higher powers of smaller primes (like 27 or 35).
Therefore, we just multiply the previous LCM (N) by 113.
∴ The correct answer is 113N.
Alternate Method
Given:
LCM(1, 2, ..., 110) = N
Calculations:
⇒ LCM(1, 2, ..., 115) = LCM(N, 111, 112, 113, 114, 115)
⇒ Let us write the prime factorization of the new terms:
⇒ 111 = 31 × 371
⇒ 112 = 24 × 71
⇒ 113 = 1131 (Prime)
⇒ 114 = 21 × 31 × 191
⇒ 115 = 51 × 231
⇒ N already contains the highest powers of all primes ≤ 110.
⇒ Max power of 3 in N is 34 = 81 and 37 is 371 → 111 completely divides N.
⇒ Max power of 2 in N is 26 = 64 and 7 is 72 = 49 → 112 completely divides N.
⇒ Max powers of 19 and 23 are 191 and 231 → 114 and 115 completely divide N.
⇒ 113 is a prime number strictly greater than 110, so it is not present as a factor in N.
⇒ Required LCM = N × 113 = 113N
∴ The correct answer is 113N.
Additional Information
LCM of Consecutive Numbers
The LCM of the first n natural numbers is formed by multiplying the highest power of every prime number that is less than or equal to n.
Extending LCM to Larger Ranges
To find the LCM of the first n+k numbers when the LCM of the first n numbers is known, simply multiply the old LCM by any prime numbers or new higher powers of primes that occur within the range (n, n+k].
Identifying New Prime Powers
If a composite number added to the range is made strictly of prime powers already smaller than n, it will not change the overall LCM. Only new prime numbers (like 113) or prime powers exceeding the previous maximums will affect it.