Why are all 4-digit Fredholl numbers composite?

1 answer:

6 0

Fredholl numbers are number, which have 2 different digits, equal number of each one, and zero cannot be the first digit. Some examples of fredholl numbers are: 355533 and 4141.
233233, 535351 or 055050 are not fredholl numbers.

In case of 4 digit numbers let's look at the possible forms of Fredholl numbers:

They have to be one of the following: xxyy, xyxy, xyyx
We can write above numbers as:
$xxyy= x*1100 + y*11 = x*11*100 + y*11 = 11 (100x+y) xyxy= x*1010 + y*101 = x*101*10 + y*101 = 101 (10x+y) xyyx=x*1001 + y*110 = x*11*91 + y*11*10 = 11 (91x+10y)$

All 4-digit Fredholl numbers are composite because they will be divisible by either 11 or 101.

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