Divisibility Formula for Numbers Ending in 5  - Proof

Niranjan Ramakrishnan
July 9, 2018

The previous post gave an empirical formula to determine if a number was divisible by a divisor ending in 5. This post offers a proof for the same.

Proposition:
Any number (10r+u) | (10x+5), if (r-x) | n for u=5, or r | n for u=0, where n=(10x+5)/5.

Proof:
If n=(10x+5)/5, n= (2x+1).

Since we're only considering dividends ending in 0 or 5, it is sufficient to show that ((10r+u) / 5) | n.

When u=5, (10r+u) /5 = (2r + 1).
(2r+1) can be rewritten as (2r -2x +2x + 1).
Thus if (r-x) |  n, then (2r + 1) | n, because n=2x+1.
QED.

When u=0, (10r+u) / 5 = 2r.
If r | n, then 2r | n.
QED.