Prime factorisation

Two-digit primes from given digits ●○○○○ 10 ptsHow many two-digit prime numbers can be formed by choosing two different digits from the set $\{2, 7, 8, 9\}$ to be used as the tens digit and units digit?
Distinct primes in 20 factorial ●○○○○ 10 ptsHow many different primes appear in the prime factorization of $20!$ (20 factorial)? Recall that $n!$ is the product of the integers from 1 to $n$ .
GCD of two prime-factored powers ●○○○○ 10 ptsCompute $\gcd(6^2\cdot 10^2,\,15^4)$ .
Power of a prime with prime shift ●●○○○ 20 ptsLet $p$ be a prime number such that $p + 3$ is also prime. Find $p^{11} - 52$ .
Primes from up to two digits ●●○○○ 20 ptsPick one or two different digits from the set $\{1, 3, 6, 7\}$ and arrange them to form a number. How many prime numbers can be created in this manner?
Composite factors of 420 ●●○○○ 20 ptsFind the number of composite factors of $420$ .

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