#math trivia for #June25: #176 is a multiple of 11 but unlike 22, 33, 44, … doesn’t have repeated digits. Or does it?

â€” Burt Kaliski Jr. (@modulomathy) June 24, 2012

Answer: 176 would have repeated digits in a *redundant decimal representation*, where each digit can have more values than the usual 0-9. In particular, if digits can have values as high as 16 (or higher), then 176 could be expressed as the repeated digits “16” “16”. Alternatively, extending hexadecimal representation (where the values from 10-15 are rendered as the letters A-F), one could write 176 = GG. (The letter G has the value 16.)

Note that this is still base 10, not a higher base; 176 is the sum of 10*G + G = 10*16 + 16 = 176.

The standard representation 176 “hides” the repeated digit within the 7, which is formed from the lower 6 of the first 16, and the upper 1 of the second.

Every multiple of 11 could be expressed as two repeated digits base 10 if we allowed large enough symbols. After 99 would come AA, then BB, then CC, and so on.

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