Multiples with no large digits

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 × CAFEBABE_hex = 42758583546_hex.

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] Gregory Galperin and Michael Reid. Multiples Without Large Digits. The American Mathematical Monthly, Vol. 126, No. 10 (December 2019), pp. 950-951.

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