That depends on the choice distribution. Your claim is true for uniform distribution over a large enough finite range of integers. If the range is too small, y

That depends on the choice distribution. Your claim is true for uniform distribution over a large enough finite range of integers. If the range is too small, you fail. If the choice is not uniform, you may fail. There is no uniform distribution over all integers, only finite subsets. Your claim fails for other distributions. The reason this mostly works is multiplying by something ending in zero gives zero, so it is the trailing digit the most. The leading digit is pretty much Benford's Law.