practice problems pending

Q1. Compute the sum of all positive integers (n) for which lcm(1,2...n) can be written as the product of 10 distinct pairwise coprime positive integers, each less than or equal to (n).
S1.
For e.g. consider a smaller problem where we need to find 4 distinct pairwise coprime factors.

For n = 5, 
LCM(1,2,3,4,5) = 60.

60 = 1.2^2.3^1.5^1

So the factors are 1,3,4,5
Each pair is co prime.
Each factor is less than 60.

So we need first 9 primes and 1 or first 10 primes to solve this.
2,3,5,7, 11,13,17,19 23,29 are the first 10 primes.

LCM(1,2... 23) = 1. 2^4. 3^2. 5^2. 7. 11. 13. 17. 19. 23
We can see the 10 factors each pairwise co prime and <= 23
Same will happen for 24,25,26,27,28.

For 29,30 we will remove 1 to get exactly 10 factors.
That's the last.
From 31 we will have at least 11 such factors.
Answer = 23 + 24... 30 = 212

Q2.


S2.








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