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P 2 Prove 30 \ p_1 - q_1 27th Indian NMO 2012

Source: INMO

February 5, 2012
number theoryprime numbersdivisibility testsnumber theory proposed

Problem Statement

Let p1<p2<p3<p4p_1<p_2<p_3<p_4 and q1<q2<q3<q4q_1<q_2<q_3<q_4 be two sets of prime numbers, such that p4−p1=8p_4 - p_1 = 8 and q4−q1=8q_4 - q_1= 8. Suppose p1>5p_1 > 5 and q1>5q_1>5. Prove that 3030 divides p1−q1p_1 - q_1.
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