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products, (a + i) | b(b + 2016), (a + i) \nmid b, (a + i)\mid (b + 2016)

Source: 2016 Saudi Arabia Pre-TST Level 4+ 2.4

September 13, 2020
number theoryProductdividesdivisible

Problem Statement

Let nn be a given positive integer. Prove that there are infinitely many pairs of positive integers (a,b)(a, b) with a,b>na, b > n such that ∏i=12015(a+i)∣b(b+2016),∏i=12015(a+i)∤b,∏i=12015(a+i)∣(b+2016)\prod_{i=1}^{2015} (a + i) | b(b + 2016), \prod_{i=1}^{2015}(a + i) \nmid b, \prod_{i=1}^{2015} (a + i)\mid (b + 2016).
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