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Divisibility Sequence (Posted on 2025-01-30)

Show that every number in the sequence

1007, 10017, 100117, 1001117, 10011117, . . .

is divisible by 53.

No Solution Yet Submitted by Danish Ahmed Khan

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re: Another solution

Comment 4 of 4 |

(In reply to Another solution by Brian Smith)

I like the inductive proof best, but it had a small typo, losing the 10.

The proof should go as below:

a(1) = 19 * 53, a(1) is a multiple of 53

a(n) = 10 a(n-1) - 53

suppose a(n-1) = k * 53

a(n) = 10 * a(n-1) - 53 = 10 (k * 53) - 53

= (10 k -1) * 53

Since 10 k -1 is integer, a(n) is a multiple of 53

Edited on January 30, 2025, 1:56 pm

Posted by Steven Lord on 2025-01-30 12:51:39

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