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Question

Let p1<p2<p3<p4 and q1<q2<q3<q4 be two sets of prime numbers such that p4p1=8 and q4q1=8. Suppose, p1>5 and q1>5. Prove that 30 divides p1q1.

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Solution

Since p4p1=8, and no prime is even, we observe that {p1,p2,p3,p4} is a subset of {p1,p1+2,p1+4,p1+6,p1+8}
Also p1 is larger than 3
If p11 (mod 3), then p1+2andp1+8 are divisible by 3.
Hence we do not get 4 primes in the set {p1,p1+2,p1+4,p1+6,p1+8}.
p12 (mod 3) and p1+4 is not a prime
We get p2=p1+2,p3=p1+6,p4=p1+8.
Consider the remainders of p1,p1+2,p1+6,p1+8 when divided by 5
If p12 (mod 5), then p1+8 is divisible by 5 and hence is not a prime
If p13 (mod 5), then p1+2 is divisibe by 5
If p14 (mod 5), then p1+6 is divisible by 5
Hence the only possibility is p11 (mod 5).
Thus we see that p11 (mod 2),p12 (mod 3)andp11 (mod 5). We conclude that p111 (mod 30).
Similarly q111 (mod 30). It follows that 30 divides p1q1.

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