2 Proof of Theorem 1

The right-most digit of N is 5. Thus, N is composite, and if we replace any digit of N , other than the right-most digit, with a different digit, then the new number obtained will be divisible by 5 and, hence, be composite. On the other hand, if we replace the right-most digit of N by a different digit, then we obtain one of N ± j where j ∈ {1, 2, 3, 4, 5} and each of these is easily seen to be composite. Thus, N has the property that if we replace any one of its digits with an arbitrary digit, the number we obtain is composite. Constructing examples of N having the above property is simplified by the fact that changing digits (arbitrarily many) of a natural number N , other than the right-most digit, results in a number divisible by gcd(N, 10). Thus, if we allow gcd(N, 10) 6= 1, then we can easily construct examples of N with the property that replacing any one of its digits with an arbitrary digit results in a composite number.