9,801

[Math.]  The largest 4-digit square number.

 

[Math.]  (Sequence of all numbers 99…999). 

9801 = (98+01)2 = 992 and trivially, 0001 = (00+01)2 = 012 and 01+99 = 100.

 

[Math.]

(*)  332×9 = 1089×9 = 9,801 = 992 & 1089×9801 = 10,673,289 = 32672

1089…1089×9 = 9801…9801  & 1089…1089×9801…9801  = 3267…32672

 

(**) 10989×9 = 98901 & 10989×98901 = 1,086,823,089 = 329672

10989…10989×9 = 98901…98901 & 10989…10989×98901...98901 = 32967…329672

 

(***)  109989×9 = 989901 & 109989×989901 = 108,878,221,089 = 3299672

109989…109989×9 = 989901…989901 &

109989…109989×989901...989901 = 329967…3299672

 

(****)  109…989×9 = 989…901 & 109...989×989…901 = 329…9672

 109…989...109…989×9 = 989…901...989…901  &

  109…989…109…989×989…901...989…901 = 329…967…329…9672.

 

9,851 & 9,857

[Math.]  Two consecutive prime numbers 9851 and 9857 form the largest prime number formed by 2 consecutive 4-digit prime numbers in both forward and backward order: 98519857 and 98579851.

 

9,862

[Math.]  Two multiples of number 4931, 4931×2 = 9862 and 4931×7 = 34,517, use each of 9 digits 1-9 once.

 

9,867

[Math.]  98672 = 9735,7689.

 

9,870

[Math.]  A triangular number (sum of all integers from 1 to 140): 9870 = 1+2+…+139+140.

 

9,871

[Math.]  The largest prime number of 4 different digits.

 

9,876

[Math.]  The largest number of 4 different digits.

 

[Math.]

1×8+1 = 9

12×8+2 = 98

123×8+3 = 987

1234×8+4 = 9,876

12,345×8+5 = 98,765

123,456×8+6 = 987,654

1,234,567×8+7 = 9,876,543

12,345,678×8+8 = 98,765,432

123,456,789×8+9 = 987,654,321

1,234,567,890×8+90 = 9,876,543,210

      (start of a new cycle)

12,345,678,901×8+901 = 98,765,432,109

123,456,789,012×8+9,002 = 987,654,321,098

1,234,567,890,123×8+90,003 = 9,876,543,210,987

12,345,678,901,234×8+900,004 = 98,765,432,109,876

123,456,789,012,345×8+9,000,005 = 987,654,321,098,765

1,234,567,890,123,456×8+90,000,006 = 9,876,543,210,987,654

12,345,678,901,234,567×8+900,000,007 = 98,765,432,109,876,543

123,456,789,012,345,678×8+9,000,000,008 = 987,654,321,098,765,432

1,234,567,890,123,456,789×8+90000000009 = 9,876,543,210,987,654,321

12,345,678,901,234,567,890×8+ 900000000090 = 98,765,432,109,876,543,210

...  (end of cycle) ...

  

9×9+(9–2) = 88

98×9+(9–3) = 888

987×9+(9–4) = 8,888

9,876×9+(9–5) = 88,888

98,765×9+(9–6) = 888,888

987,654×9+(9–7) = 8,888,888

9,876,543×9+(9–8) = 88,888,888

98,765,432×9+(9–9) = 888,888,888

987,654,321×9+(9–10) = 8,888,888,888

9,876,543,210×9+(9–11) = 88,888,888,888

98,765,432,099×9+(9–12) = 888,888,888,888 = 98,765,432,100×9–12.

 

[Math.]  Two multiples of number 2469, 2469×4 = 9,876 and 2469×5 = 12,345, use each of 9 digits 1-9 once.

 

9,887

[Math.]  A prime number.

 

9,898

[Math.]  98982 = 97,970,404.

 

9,899

[Math.]  The reciprocal of numbers in the series 89, 9899, 998999… (i.e, of the form (102n–10n–1), for n ³ 1) contains n rightmost digit of the first nth Fibonacci numbers up to the second biggest n-digit Fibonacci number: 1/89 = 0.011235955… * 1/9899 = 0.000101020305081321345590…