Define the sequence recursively, a(n) = 10*a(n-1) - 53 with initial a(1) = 1007. 1007 = 53*19. The first term is a multiple of 53.
Now create an induction hypothesis: if a(n-1) is a multiple of 53 then a(n) is a multiple of 53. Assume a(n-1) = 53*k, so a(n) = 10*53*k - 53 = 53*(10*k-1). Therefore a(n) is a multiple of 53.
Since the first term is a multiple of 53 our induction hypothesis applies and we can subsequently say every term in the sequence is a multiple of 53. QED
Fixed typo noticed by Steven.
Edited on January 31, 2025, 11:07 am
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