## 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.

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