%I #40 Apr 19 2017 11:34:41

%S 10,15,18,20,30,40,45,50,60,70,80,90,100,200,300,400,500,600,700

%N Elastic numbers: numbers n with two or more digits such that for any number m made by inserting any number of 0's between the digits of n, m is a multiple of n.

%C 15, 18 and 45 are the only elastic numbers with no 0 as a digit. Proof: let n = 10*m+k, 0<k<10. Then n is elastic only if 90*m is a multiple of n, i.e., 90*m = n*q = (10*m+k)*q = 10*m*q + k*q. It follows that q<9, so k*q<100. But k*q = 10*m*(9-q) which is no less than 100 if n has at least 3 digits. Contradiction. - _Andrey Zabolotskiy_, Apr 18 2017

%D D. Griller, Elastic Numbers, Rational Falcon, 2017, 45 & 95.

%e 15 is an elastic number because 105, 1005, 10005, 100005, ... are all multiples of 15.

%K nonn,base

%O 1,1

%A _Matthew Scroggs_, Apr 18 2017