# Probability of the sum of n numbers giving the same last d digits

Can anybody give me a hint how to approach the following problem, please?

I actually am having a hard time stating the problem. I think an example would help you understand what the problem is.

Suppose I have 5 numbers each can be max 3 digits. I sum them together. What's the probability of the sum having the last 3 digits 500?

Example of 5 numbers having the last 3 digits 500 are (100, 100, 100, 100, 100) and (0, 0, 0, 0, 1500).

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Practical application: When the transmitter adds d check digits or b bits of checksum to a message, what is the probability that the receiver will (incorrectly) fail to detect any errors in a random message? –  David Cary Apr 26 '13 at 15:11

HINT $\$ For $\rm\ x + y + z\equiv 500\ \ (mod\ 1000)$ you have $1000$ choices for $\rm\:x\:$, $1000$ choices for $\rm\:y\:$, and then $\rm\:z\:$ is uniquely determined by the linear equation.