Here’s a curious theorem I stumbled across recently [1]. Take an integer _N_ which is not a multiple of 10. Then there is some multiple of _N_ which only contains the digits 1, 2, 3, 4, and 5. For example, my business phone number 8324228646 has a couple 8s and a couple 6s. But 6312 × 8324228646 = 52542531213552 which contains only digits 1 through 5. For a general base _b_ , let _p_ be the smallest prime factor of _b_. Then for every integer _N_ that is not a multiple of _b_ , there is some multiple of _N_ whose base _b_ representation contains only the digits 1, 2, 3, …, _b_ /_p_. This means that for every number _N_ that is not a multiple of 16, there is some _k_ such that the hex representation of _kN_ contains only the digits 1 through 8. For example, if we take the magic number at the beginning of every Java class file, 0xCAFEBABE, we find 1341 × CAFEBABEhex = 42758583546hex. In the examples above, we’re looking for multiple containing only half the possible digits. If the largest prime dividing the base is larger than 2 then we can find a multiples with digits in a smaller range. For example, in base 35 we can find a multiple containing only the digits 1 through 7. [1] Daniel Sitaru and Leonard Giugiuc. Sum of powers of the sides of a triangle. The American Mathematical Monthly, Vol. 126, No. 2 (February 2019), p. 188

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