2021/102) What is the largest natural number n below 50 such that LCM (n, n + 1, . . . , 50) = LCM (1, 2, . . . , 50), where LCM stands for least common multiple.?

 it is 27. because if we leave out 27 then none of the numbers above 27 is divisible by 27 so we shall not have a factor 27 of the LCM but for numbers 1 to 50 LCM shall have a factor 27.

27 is not by magic. it has to a prime number or a power of a prime. if it is composite other than power of a prime then is has got smaller factor and this is taken care of because smaller numbers are taken care of. so we should look for a number greater than 1/2 of the number because if a is taken care of the 2a is taken care of. in LCM. so we look through numbers 26,27 etc and find 27.