111...000...???
Number Theory
Level
2
n be any natural number. Is it always true that n has a (nonzero) multiple whose representation (base 10) contains only zeroes and ones?
Never true
Depends on value of n
Always true
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1 solution
We employ the famous and useful "pigeon-hole principle": If there are more pigeons than holes, then some hole must contain at least two pigeons. There are only n numbers
modulo n, but the set {1,11, 111, 1111,... }, whose largest member has n+1 digits, has size n + 1; thus, it contains two numbers whose values are equal modulo n. Subtract one from the other!