109,999,999,999,989 & 329,999,999,999,967 & 989,999,999,999,901

109,891,098,910,989 & 329,673,296,732,967 & 989,019,890,198,901

[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

 ...

(***)   109,999,999,999,989×9 = 989,999,999,999,901 &

109,999,999,999,989×989,999,999,999,901 = 329,999,999,999,9672

1099999999989…109999999989×9 = 9899999999901…9899999999901 &

1099999999989…1099999999989×9899999999901...9899999999901 =

                       = 32999999967…32999999999672

 ...

(****)  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.

 

109,876,543,456,789

[Math.]

99,889×11+(11–1) = 1,098,789

998,877,889×11+(11–1) = 10,987,656,789

9,988,776,677,889×11+(11–1) = 109,876,543,456,789

99,887,766,556,677,889×11+(11–1) = 1,098,765,432,123,456,789

 

111,111,111,111,111 & 123,456,787,654,321

[Math.]

1×1 = 1

11×11 = 121

111×111 = 12321

1111×1111 = 1234321

11111×11111 = 123454321

111111×111111 = 12345654321         

1111111×1111111 = 1234567654321

11111111×11111111 = 123456787654321

111111111×111111111 = 12345678987654321

1111111111×1111111111 = 1234567900987654321

11111111111×11111111111 = 123456790120987654321

 

111,111,111,111,111 & 333,333,333,333,333

[Math.]

3×37 = 111

33×3367 = 111,111

333×333667 = 111,111,111

3333×33336667 = 111,111,111,111

33333×3333366667 = 111,111,111,111,111

333333×333333666667 = 111,111,111,111,111,111

3333333×33333336666667 = 111,111,111,111,111,111,111

33333333×3333333366666667 = 111,111,111,111,111,111,111,111

333333333×333333333666666667 = 111,111,111,111,111,111,111,111,111

3333333333×33333333336666666667 = 111,111,111,111,111,111,111,111,111,111

33333333333×3333333333366666666667 =

= 111,111,111,111,111,111,111,111,111,111,111

333333333333×333333333333666666666667 =

= 111,111,111,111,111,111,111,111,111,111,111,111

3333333333333×33333333333336666666666667 =

= 111,111,111,111,111,111,111,111,111,111,111,111,111

33,333,333,333,333×3333333333333366666666666667 =

= 111,111,111,111,111,111,111,111,111,111,111,111,111,111

333,333,333,333,333×333333333333333666666666666667 =

= 111,111,111,111,111,111,111,111,111,111,111,111,111,111,111

...

 

123,456,787,654,321

[Math.]

11×11 = 121

112,211×11 = 1,234,321

1,122,332,211×11 = 12,345,654,321

11,223,344,332,211×11 = 123,456,787,654,321

112,233,445,544,332,211×11 = 1,234,567,900,987,654,321        

 

[Math.]  

121×(1+2+1) = 484 = 222

12,321×(1+2+3+2+1) = 110,889 = 3332

1,234,321×(1+2+3+4+3+2+1) = 19,749,136 = 4,4442

123,454,321×(1+2+3+4+5+4+3+2+1) = 3,086,358,025 = 55,5552

12,345,654,321×(1+2+3+4+5+6+5+4+3+2+1) = 444,443,555,556 = 666,6662

1,234,567,654,321×(1+2+3+4+5+6+7+6+5+4+3+2+1) = 

                                                                         =   60,493,815,061,729 = 7,777,7772

123,456,787,654,321×(1+2+3+4+5+6+7+8+7+6+5+4+3+2+1) =

                                                                     = 7,901,234,409,876,544 = 88,888,8882

    12,345,678,987,654,321×(1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1) =

                                                              = 999,999,998,000,000,001 = 999,999,9992.

  

123,456,789,012,345 & 987,654,321,098,765

[Math.] 

1×8+1 = 9

12×8+2 = 98

123×8+3 = 987   

1234×8+4 = 9876

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 new cycle)

12,345,678,901×8+901 = 98,765,432,109

123,456,789,012×8+9002 = 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+90,000,000,009 = 9,876,543,210,987,654,321

12,345,678,901,234,567,890×8+ 900,000,000,090 = 98,765,432,109,876,543,210

  (end of cycle) 

 

142,857,142,857,142

[Math.]

1×7 + 3 = 10

14×7 + 2 = 100

142×7 + 6 = 1000

1428×7 + 4 = 10000

14285×7 + 5 = 100000

142857×7 + 1 = 1000000

 

1428571×7 + 3 = 10000000

14285714×7 + 2 = 100000000

142857142×7 + 6 = 1000000000

1428571428×7 + 4 = 10000000000

14285714285×7 + 5 = 100000000000

142857142857×7 + 1 = 1000000000000

 

1428571428571×7 + 3 = 10000000000000

14285714285714×7 + 2 = 100000000000000

142857142857142×7 + 6 = 1000000000000000

1428571428571428×7 + 4 = 10000000000000000

14285714285714285×7 + 5 = 100000000000000000

142857142857142857×7 + 1 = 1000000000000000000

 

 

159,999,999,999,984 & 279,999,999,999,972 & 489,999,999,999,951 &

783,998,432,000,784

[Math.]

7,683,984 = 1584×4851 = 27722                                                                    

782,432,784 = 15984×48951 = 279722

78,384,320,784 = 159984×489951 = 2799722

7,839,843,200,784 = 1599984×4899951 = 27999722

783,998,432,000,784 = 15999984×48999951 = 279999722

78,399,984,320,000,784 = 159999984×489999951 = 2799999722

7,839,999,843,200,000,784 = 1599999984×4899999951 = 27999999722

783,999,998,432,000,000,784 = 15999999984×48999999951 = 279999999722

78,399,999,984,320,000,000,784 = 159999999984×489999999951 = 2799999999722

 7839…9984320000784 = 159…999984×489…999951 = 279…9999722.

 

219,999,999,999,978 & 439,999,999,999,956 & 879,999,999,999,912 

[Math.] 

(*)   8712 = 2178×4 & 8712×2178 = 18,974,736 = 43562

87128712 = 21782178×4 & 87128712×21782178 = 435643562

 

(**)   87912 = 21978×4 & 21978×87912 = 1,932,129,936 = 439562

8791287912 = 21978…21978×4 & 21978…21978×87912…87912 = 43956…439562

...

(***)   879,999,999,999,912 = 219,999,999,999,978×4 & 

2199,999,999,999,978×87,999,999,999,912 = 439,999,999,999,9562

8799...99128799...9912 = 2199...99782199...9978×4 &

2199...99782199...9978×8799...99128799...9912 = 4399...99564399...99562

 ...

 (****)   879…912 = 219…978×4 & 219…978×879…912 = 439…9562

879…912879…912 = 219…978…219…978×4 &

219…978219…978×879...912…879...912 = 439…956439…9562.

 

109,999,999,999,989 & 329,999,999,999,967 & 989,999,999,999,901

109,891,098,910,989 & 329,673,296,732,967 & 989,019,890,198,901

[Math.] (Above)

 

299,999,999,999,996

[Math.]  

296×333,667 = 98,765,432

2,996×33,336,667 = 99,876,654,332

29,996×3,333,366,667 = 99,987,666,543,332

299,996×333,333,666,667 = 99,998,766,665,433,332

2,999,996×33,333,336,666,667 = 99,999,876,666,654,333,332

29,999,996×3,333,333,366,666,667 = 99,999,987,666,666,543,333,332

299,999,996×333,333,333,666,666,667 = 99,999,998,766,666,665,433,333,332

2,999,999,996×33,333,333,336,666,666,667 = 

                              = 99,999,999,876,666,666,654,333,333,332 

29,999,999,996×3,333,333,333,366,666,666,667 =

= 99,999,999,998,766,666,666,663,333,333,332

299,999,999,996×333,333,333,333,366,666,666,667 =

= 99,999,999,998,766,666,666,665,433,333,333,332

...

 

333,333,333,333,333 & 111,111,111,111,111

[Math.]

3×37 = 111

33×3367 = 111,111

333×333667 = 111,111,111

3333×33336667 = 111,111,111,111

33333×3333366667 = 111,111,111,111,111

333333×333333666667 = 111,111,111,111,111,111

3333333×33333336666667 = 111,111,111,111,111,111,111

33333333×3333333366666667 = 111,111,111,111,111,111,111,111

333333333×333333333666666667 = 111,111,111,111,111,111,111,111,111

3333333333×33333333336666666667 = 111,111,111,111,111,111,111,111,111,111

33333333333×3333333333366666666667 =

= 111,111,111,111,111,111,111,111,111,111,111

333333333333×333333333333666666666667 =

= 111,111,111,111,111,111,111,111,111,111,111,111

3333333333333×33333333333336666666666667 =

= 111,111,111,111,111,111,111,111,111,111,111,111,111

33,333,333,333,333×3333333333333366666666666667 =

= 111,111,111,111,111,111,111,111,111,111,111,111,111,111

333,333,333,333,333×333333333333333666666666666667 =

= 111,111,111,111,111,111,111,111,111,111,111,111,111,111,111

...

 

369,999,999,999,999 & 333,666,699,999,999

[Math.]

3692 = 136,161

36992 = 13682601

36,9992 = 1368926001                                                  

369,9992 = 136899260001

3,699,9992 = 13689992600001

36,999,9992 = 1368999926000001

369,999,9992 = 136899999260000001

3,699,999,9992 = 13689999992600000001                          

36,999,999,9992 = 1368999999926000000001

369,999,999,9992 = 136899999999260000000001

   

3366992 = 113366,216601

336699992 = 11336688,32660001

33669999992 = 1133668899,3266000001                        

3366999999992 = 113366889999,326600000001

333,666,9992 = 111,333,666,221,666,001

333,666,999,9992 = 111,333,666,888,332,666,000,001

333,666,999,999,9992 = 111,333,666,888,999,332,666,000,000,001

3333,6666,99992 = 1111,3333,6666,2221,6666,0001

3333,6666,9999,99992 = 1111,3333,6666,8888,3332,6666,0000,0001


666,666,666,666,666.3333333333333332…

[Math.]  The square root of the number formed by n digits 4 followed by n digits 0 starts with n digits 6, decimal point, and n digits 3:

401/2 = 6.32…

44001/2 = 66.332…

4440001/2 = 666.3332…

444400001/2 = 6666.33332…

44444000001/2 = 66666.333332…

4444440000001/2 = 666666.3333332…

444444400000001/2 = 6666666.33333332…

44444444000000001/2 = 66666666.333333332…

4444444440000000001/2 = 666666666.3333333332…

44444444440000000001/2 = 6666666666.33333333332…

4444444444400000000001/2 = 66666666666.333333333332…

444444444444000000000001/2 = 666666666666.3333333333332…

44444444444440000000000001/2 = 6666666666666.33333333333332…

4444444444444400000000000001/2 = 66666666666666.333333333333332…

444444444444444000000000000001/2 = 666666666666666.3333333333333332…

44444444444444440000000000000001/2 = 6666666666666666.33333333333333332…


888,888,888,888,888 & 987,654,320,987,655

[Math.]

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.

987,654,320,988×9+(9–13) = 8,888,888,888,888

9,876,543,209,877×9+(9–14) = 88,888,888,888,888

98,765,432,098,766×9+(9–15) = 888,888,888,888,888

987,654,320,987,655×9+(9–16) = 8,888,888,888,888,888

9,876,543,209,876,544×9+(9–17) = 88,888,888,888,888,888

...


109,999,999,999,989 & 329,999,999,999,967 & 989,999,999,999,901

109,891,098,910,989 & 329,673,296,732,967 & 989,019,890,198,901

[Math.] (See more above)