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Suppose a and b are positive integers such that gcd(a,b) is divisible by exactly 7 distinct primes and lcm[a,b] is divisible by exactly 10 distinct primes.  If a has fewer distinct prime factors than b, then a has at most how many distinct prime factors?

 Nov 16, 2020
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GCD[17!,  29!] =17!

17! = 2^15 * 3^6 * 5^3 * 7^2 * 11 * 13 * 17 [It has 7 distinct prime factors]

 

LCM[17!,  29!] =29!

29! = 2^25 * 3^13 * 5^6 * 7^4 * 11^2 * 13^2 * 17 * 19 * 23 * 29 [It has 10 distinct prime factors]

 

a=has at most 7 distinct prime factors.

 Nov 16, 2020

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