Question

sum of 30 not necessarily distinct positive integers is901 LCM of these 30 numbers is​

Answer 1

Answer:

We show that the least possible LCM is 42.

Let the 20 (positive) integers be a1, a2, a3, …, a20, with 1≤a1≤a2≤a3≤⋯≤a20. Since 801>20×40, at least one of the integers must exceed 40. So a20≥41, and consequently their lcm ≥41.

If the lcm of these integers is to equal 41, then each ai must equal either 1 or 41. Since not all ai can equal 41 (41∤801), there must exist k∈{1,2,3,…,19} such that ak=1 and ak+1=41. Since 779=41×19 is the largest multiple of 41 less than 801, and 801=(41×19)+22, no such k exists.

Hence the smallest possible lcm is 42.

The largest multiple of 42 less than or equal to 801 is 798=42×19. If we choose a1=3 and ai=42 for 2≤i≤20, then we find that the lcm equals 42.

This proves our claim. ■