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  1. It is a four-digit number.
  2. Its largest place value is its one place.
  3. The squares of each of its digits are less than nine.
  4. Its thousandths didgit divided by its ones digits is 50%.
  5. The sum of two pf its digits is 0.
  6. The difference between its ones digits and its thousands digit is 1.
  7. The sum of all of its digits is 3.
  8. It has only one prime digit.
  9. It is greater than 2.
  10. Its tenths digit is 0.
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  • 1
    $\begingroup$ What have you done? Were are you stuck? $\endgroup$ – Pragabhava Nov 12 '12 at 17:57
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Item 4 is stated incorrectly: the quotient of two digits can be $\frac12$, but it can’t be $50$%.

The first two items tell us that the number has the form $a.bcd$, and the third tells us that $a,b,c,d\in\{0,1,2\}$. Item 4 says that $a=2d$. Item 5 says that two of the digits are $0$. Item 6 says that $|a-d|=1$; since $a=2d$, we must have $a=2$ and $d=1$, so the number is $2.001$. The remaining conditions are superfluous.

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  • $\begingroup$ Small point: I'm reading that rule 3 says $a$, $b$, $c$, and $d$ cannot be $3$ because $3^2\nless 9$. Also, I've long read "fifty percent" to be like "fifty miles per hour" as in miles/hours or in the case of "50%" I read $50/100$ (literally per cent) and therefore 50% could be seen as an alternate way to denote $1/2$. $\endgroup$ – Todd Wilcox Nov 12 '12 at 18:05
  • $\begingroup$ @Todd: You’re right about Item 3; I read too quickly and registered it as $\le 9$. I don’t consider $50$% an acceptable substitute for $\frac12$ in this context: the result of dividing one integer by another is not a percentage unless the context calls for one. $\endgroup$ – Brian M. Scott Nov 12 '12 at 18:13

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