A316. SVP 7 ou plus, rien d'autre
Démontrer qu’il existe une infinité de couples d’entiers naturels tels que chacun d’eux et leur produit contiennent exclusivement des chiffres supérieurs ou égaux à 7.
1. Un (premier) couple de base et la démonstration Soient ܽ
et ܾ
ܽ
= 8. 10
ିଵ+ 7
9 ሺ10
ିଵ− 1ሻ
ܾ
= 8. 10
ିଵ+ ሺ10
ିଷ− 1ሻ. 10
ଶ+ 87
ܽ
s’écrit avec le chiffre 8 suivi de ݊ − 1 chiffres 7 ( ܽ
= 877777 )
ܾ
s’écrit avec le chiffre 8 suivi de ݊ − 3 chiffres 9 puis 8 et 7 ( ܾ
= 899987 ) On va montrer que le produit ܽ
. ܾ
= ܿ
s’écrit avec seulement de 7, des 8 et des 9.
ܿ
= ܽ
. ܾ
ሺ8. 10ିଵሻ. ሺ8. 10ିଵሻ +ሺ8. 10ିଵሻ. ሺ10ିଷ− 1ሻ. 10ଶ +ሺ8. 10ିଵሻ. 87 +7
9 ሺ10ିଵ− 1ሻ. 8. 10ିଵ +7
9 ሺ10ିଵ− 1ሻ. ሺ10ିଷ− 1ሻ. 10ଶ +7
9 ሺ10ିଵ− 1ሻ. 87 +64. 10ଶିଶ +8. 10ଶିଶ− 8. 10ାଵ +696. 10ିଵ +56
9 10ଶିଶ−56
9 10ିଵ +7
9 10ଶିଶ−7
9 10ାଵ−7
9 10ିଵ+7
9 10ଶ +609
9 10ିଵ−609 9
Termes en 2n-2 Termes en n+1 Termes en n-1 Termes constants
64. 10ଶିଶ+ 8. 10ଶିଶ+56
9 10ଶିଶ+7
9 10ଶିଶ −8. 10ାଵ−7 9 10ାଵ
+696. 10ିଵ−56
9 10ିଵ−7 9 10ିଵ +609
9 10ିଵ
+7
9 10ଶ−609 9 ൬64 + 8 +56
9 + 7
9൰ . 10ଶିଶ ൬−8 −7
9൰ 10ାଵ ൬696 −56 9 −
7 9 +
609 9 ൰ 10ିଵ
91 9
79. 10ଶିଶ −79
9 10ାଵ
6810 9 10ିଵ 79. 10ଶିଶ−10ଶିଶ +10ଶିଶ−79
9 10ାଵ
681 9 10
78. 10ଶିଶ ൬10ିସ−79
90൰ 10ାଶ ൬10ିସ−79
90−11
90൰ 10ାଶ 681 9 10+11
90 10ାଶ
ሺ10ିସ− 1ሻ10ାଶ 791
9 10 791
9 10−8
9 10 +8
9 10
87.10 +8
9 10−800
9 +800
9 +91 9 +800
9 10ିଶ−800 8 9
9 ሺ10ିଶ− 1ሻ10ଶ 99
ܿ
= ܽ
. ܾ
= 78. 10
ଶିଶ+ ሺ10
ିସ− 1ሻ10
ାଶ+ 87.10
+ 8
9 ሺ10
ିଶ− 1ሻ10
ଶ+ 99
ܿ
commence par 78, suivi de ݊ − 4 chiffres 9, puis 87, suivi de ݊ − 2 chiffres 8, et se termine par 99.
ܿ
= 789 987 888 899
݊
ܽ ܾܿ
= ܽ
. ܾ
3 877 887 777899
4 8777 8987 78878899
5 87777 89987 7898788899
6 877777 899987 789987888899
7 8777777 8999987 78999878888899 8 87777777 89999987 7899998788888899 9 877777777 899999987 789999987888888899 10 8777777777 8999999987 78999999878888888899
…
2. Le deuxième un peu différent avec ܽ
= constante
ܽ
= 8877
ܾ
= 8887 10
ସ− 1 10
ସ− 1
ܿ
= 7 889. 10
ସ+ 7 788 10
ସሺିଶሻ− 1
10
ସ− 1 10
଼+ 77 879 899 et on a bien ∶ ܽ
. ܾ
= ܿ
1 8 877 8887 7888 9899 ok
2 8 877 8887 8887 7889 7787 9899 ok
3 8 877 8887 8887 8887 7889 7788 7787 9899 ok
4 8 877 8887 8887 8887 8887 7889 7788 7788 7787 9899 ok
5 8 877 8887 8887 8887 8887 8887 7889 7788 7788 7788 7787 9899 ok
6 8 877 8887 8887 8887 8887 8887 8887 7889 7788 7788 7788 7788 7787 9899 ok
7 8 877 8887 8887 8887 8887 8887 8887 8887 7889 7788 7788 7788 7788 7788 7787 9899 ok
8 8 877 8887 8887 8887 8887 8887 8887 8887 8887 7889 7788 7788 7788 7788 7788 7788 7787 9899 ok
9 8 877 8887 8887 8887 8887 8887 8887 8887 8887 8887 7889 7788 7788 7788 7788 7788 7788 7788 7787 9899 ok 10 8 877 8887 8887 8887 8887 8887 8887 8887 8887 8887 8887 7889 7788 7788 7788 7788 7788 7788 7788 7788 7787 9899 ok 11 8 877 8887 8887 8887 8887 8887 8887 8887 8887 8887 8887 8887 7889 7788 7788 7788 7788 7788 7788 7788 7788 7788 7787 9899 ok 12 8 877 8887 8887 8887 8887 8887 8887 8887 8887 8887 8887 8887 8887 7889 7788 7788 7788 7788 7788 7788 7788 7788 7788 7788 7787 9899 ok 13 8 877 8887 8887 8887 8887 8887 8887 8887 8887 8887 8887 8887 8887 8887 7889 7788 7788 7788 7788 7788 7788 7788 7788 7788 7788 7788 7787 9899 ok
3. La démonstration, ainsi que celles des couples suivants, doit être identique à celle déjà effectuée.
ݑݎ ݊ ≥ 3
൬89 ሺ10− 1ሻ. 10 + 7൰൫ሺ10ିଷ− 1ሻ. 10ସ+ 8877൯
= 8
9 ሺ10
ିଷ− 1ሻ. 10
ାହ+ 7888. 10
ାଵ+ 7
9 ሺ10
ିଷ− 1ሻ. 10
ସ+ 9899
3 8 887 8 877 78 889 899 ok
4 88 887 98 877 8 788 879 899 ok
5 888 887 998 877 887 888 779 899 ok
6 8 888 887 9 998 877 88 878 887 779 899 ok
7 88 888 887 99 998 877 8 888 788 877 779 899 ok
8 888 888 887 999 998 877 888 887 888 777 779 899 ok
9 8 888 888 887 9 999 998 877 88 888 878 887 777 779 899 ok
10 88 888 888 887 99 999 998 877 8 888 888 788 877 777 779 899 ok
11 888 888 888 887 999 999 998 877 888 888 887 888 777 777 779 899 ok
12 8 888 888 888 887 9 999 999 998 877 88 888 888 878 887 777 777 779 899 ok
13 88 888 888 888 887 99 999 999 998 877 8 888 888 888 788 877 777 777 779 899 ok
14 888 888 888 888 887 999 999 999 998 877 888 888 888 887 888 777 777 777 779 899 ok
15 8 888 888 888 888 887 9 999 999 999 998 877 88 888 888 888 878 887 777 777 777 779 899 ok
16 88 888 888 888 888 887 99 999 999 999 998 877 8 888 888 888 888 788 877 777 777 777 779 899 ok 17 888 888 888 888 888 887 999 999 999 999 998 877 888 888 888 888 887 888 777 777 777 777 779 899 ok 18 8 888 888 888 888 888 887 9 999 999 999 999 998 877 88 888 888 888 888 878 887 777 777 777 777 779 899 ok 19 88 888 888 888 888 888 887 99 999 999 999 999 998 877 8 888 888 888 888 888 788 877 777 777 777 777 779 899 ok 20 888 888 888 888 888 888 887 999 999 999 999 999 998 877 888 888 888 888 888 887 888 777 777 777 777 777 779 899 ok 21 8 888 888 888 888 888 888 887 9 999 999 999 999 999 998 877 88 888 888 888 888 888 878 887 777 777 777 777 777 779 899 ok 22 88 888 888 888 888 888 888 887 99 999 999 999 999 999 998 877 8 888 888 888 888 888 888 788 877 777 777 777 777 777 779 899 ok 23 888 888 888 888 888 888 888 887 999 999 999 999 999 999 998 877 888 888 888 888 888 888 887 888 777 777 777 777 777 777 779 899 ok 24 8 888 888 888 888 888 888 888 887 9 999 999 999 999 999 999 998 877 88 888 888 888 888 888 888 878 887 777 777 777 777 777 777 779 899 ok 25 88 888 888 888 888 888 888 888 887 99 999 999 999 999 999 999 998 877 8 888 888 888 888 888 888 888 788 877 777 777 777 777 777 777 779 899 ok
4.
ݑݎ ݊ ≥ 6
൬89 ሺ10ିସ− 1ሻ. 10ସ+ 7987൰൫ሺ10ିଷ− 1ሻ. 10ଷ+ 777൯
= 8
9 ሺ10
ିସ− 1ሻ. 10
ାସ+ 7788. 10
+ 7
9 ሺ10
ି− 1ሻ. 10
+ 978899
6 887 987 999 777 887 788 978 899 ok
7 8 887 987 9 999 777 88 877 887 978 899 ok
8 88 887 987 99 999 777 8 888 778 877 978 899 ok
9 888 887 987 999 999 777 888 887 788 777 978 899 ok
10 8 888 887 987 9 999 999 777 88 888 877 887 777 978 899 ok
11 88 888 887 987 99 999 999 777 8 888 888 778 877 777 978 899 ok
12 888 888 887 987 999 999 999 777 888 888 887 788 777 777 978 899 ok
13 8 888 888 887 987 9 999 999 999 777 88 888 888 877 887 777 777 978 899 ok
14 88 888 888 887 987 99 999 999 999 777 8 888 888 888 778 877 777 777 978 899 ok
15 888 888 888 887 987 999 999 999 999 777 888 888 888 887 788 777 777 777 978 899 ok
16 8 888 888 888 887 987 9 999 999 999 999 777 88 888 888 888 877 887 777 777 777 978 899 ok
17 88 888 888 888 887 987 99 999 999 999 999 777 8 888 888 888 888 778 877 777 777 777 978 899 ok 18 888 888 888 888 887 987 999 999 999 999 999 777 888 888 888 888 887 788 777 777 777 777 978 899 ok 19 8 888 888 888 888 887 987 9 999 999 999 999 999 777 88 888 888 888 888 877 887 777 777 777 777 978 899 ok 20 88 888 888 888 888 887 987 99 999 999 999 999 999 777 8 888 888 888 888 888 778 877 777 777 777 777 978 899 ok 21 888 888 888 888 888 887 987 999 999 999 999 999 999 777 888 888 888 888 888 887 788 777 777 777 777 777 978 899 ok 22 8 888 888 888 888 888 887 987 9 999 999 999 999 999 999 777 88 888 888 888 888 888 877 887 777 777 777 777 777 978 899 ok 23 88 888 888 888 888 888 887 987 99 999 999 999 999 999 999 777 8 888 888 888 888 888 888 778 877 777 777 777 777 777 978 899 ok 24 888 888 888 888 888 888 887 987 999 999 999 999 999 999 999 777 888 888 888 888 888 888 887 788 777 777 777 777 777 777 978 899 ok 25 8 888 888 888 888 888 888 887 987 9 999 999 999 999 999 999 999 777 88 888 888 888 888 888 888 877 887 777 777 777 777 777 777 978 899 ok
5.
ݑݎ ݊ ≥ 7
൬89 ሺ10ିଶ− 1ሻ. 10ଶ+ 77൰൫ሺ10ି− 1ሻ. 10+ 898887൯
= 8
9 ሺ10
ି− 1ሻ. 10
ା+ 798998. 10
+ 7
9 ሺ10
ି− 1ሻ. 10
+ 8979899
7 8 888 877 9 898 887 87 989 988 979 899 ok
8 88 888 877 99 898 887 8 879 899 878 979 899 ok
9 888 888 877 999 898 887 888 798 998 778 979 899 ok
10 8 888 888 877 9 999 898 887 88 887 989 987 778 979 899 ok
11 88 888 888 877 99 999 898 887 8 888 879 899 877 778 979 899 ok
12 888 888 888 877 999 999 898 887 888 888 798 998 777 778 979 899 ok
13 8 888 888 888 877 9 999 999 898 887 88 888 887 989 987 777 778 979 899 ok
14 88 888 888 888 877 99 999 999 898 887 8 888 888 879 899 877 777 778 979 899 ok
15 888 888 888 888 877 999 999 999 898 887 888 888 888 798 998 777 777 778 979 899 ok
16 8 888 888 888 888 877 9 999 999 999 898 887 88 888 888 887 989 987 777 777 778 979 899 ok
17 88 888 888 888 888 877 99 999 999 999 898 887 8 888 888 888 879 899 877 777 777 778 979 899 ok 18 888 888 888 888 888 877 999 999 999 999 898 887 888 888 888 888 798 998 777 777 777 778 979 899 ok 19 8 888 888 888 888 888 877 9 999 999 999 999 898 887 88 888 888 888 887 989 987 777 777 777 778 979 899 ok 20 88 888 888 888 888 888 877 99 999 999 999 999 898 887 8 888 888 888 888 879 899 877 777 777 777 778 979 899 ok 21 888 888 888 888 888 888 877 999 999 999 999 999 898 887 888 888 888 888 888 798 998 777 777 777 777 778 979 899 ok 22 8 888 888 888 888 888 888 877 9 999 999 999 999 999 898 887 88 888 888 888 888 887 989 987 777 777 777 777 778 979 899 ok 23 88 888 888 888 888 888 888 877 99 999 999 999 999 999 898 887 8 888 888 888 888 888 879 899 877 777 777 777 777 778 979 899 ok 24 888 888 888 888 888 888 888 877 999 999 999 999 999 999 898 887 888 888 888 888 888 888 798 998 777 777 777 777 777 778 979 899 ok 25 8 888 888 888 888 888 888 888 877 9 999 999 999 999 999 999 898 887 88 888 888 888 888 888 887 989 987 777 777 777 777 777 778 979 899 ok
ݑݎ ݊ ≥ 6
ቆ78. 10ିଶ+7
9 ሺ10ିଶ− 1ሻቇሺ98. 10ିଶ+ ሺ10ିସ− 1ሻ. 10ଶ+ 87ሻ= 7798. 10ଶିସ+ ሺ10ି− 1ሻ. 10ାଶ+ 889. 10ିଵ+8
9 ሺ10ିଷ− 1ሻ. 10ଶ+ 99
6 787 777 989 987 779 888 988 899 ok
7 7 877 777 9 899 987 77 989 889 888 899 ok
8 78 777 777 98 999 987 7 798 998 898 888 899 ok
9 787 777 777 989 999 987 779 899 988 988 888 899 ok
10 7 877 777 777 9 899 999 987 77 989 999 889 888 888 899 ok
11 78 777 777 777 98 999 999 987 7 798 999 998 898 888 888 899 ok
12 787 777 777 777 989 999 999 987 779 899 999 988 988 888 888 899 ok
13 7 877 777 777 777 9 899 999 999 987 77 989 999 999 889 888 888 888 899 ok
14 78 777 777 777 777 98 999 999 999 987 7 798 999 999 998 898 888 888 888 899 ok
15 787 777 777 777 777 989 999 999 999 987 779 899 999 999 988 988 888 888 888 899 ok
16 7 877 777 777 777 777 9 899 999 999 999 987 77 989 999 999 999 889 888 888 888 888 899 ok
17 78 777 777 777 777 777 98 999 999 999 999 987 7 798 999 999 999 998 898 888 888 888 888 899 ok 18 787 777 777 777 777 777 989 999 999 999 999 987 779 899 999 999 999 988 988 888 888 888 888 899 ok 19 7 877 777 777 777 777 777 9 899 999 999 999 999 987 77 989 999 999 999 999 889 888 888 888 888 888 899 ok 20 78 777 777 777 777 777 777 98 999 999 999 999 999 987 7 798 999 999 999 999 998 898 888 888 888 888 888 899 ok 21 787 777 777 777 777 777 777 989 999 999 999 999 999 987 779 899 999 999 999 999 988 988 888 888 888 888 888 899 ok 22 7 877 777 777 777 777 777 777 9 899 999 999 999 999 999 987 77 989 999 999 999 999 999 889 888 888 888 888 888 888 899 ok 23 78 777 777 777 777 777 777 777 98 999 999 999 999 999 999 987 7 798 999 999 999 999 999 998 898 888 888 888 888 888 888 899 ok 24 787 777 777 777 777 777 777 777 989 999 999 999 999 999 999 987 779 899 999 999 999 999 999 988 988 888 888 888 888 888 888 899 ok 25 7 877 777 777 777 777 777 777 777 9 899 999 999 999 999 999 999 987 77 989 999 999 999 999 999 999 889 888 888 888 888 888 888 888 899 ok
7.
ݑݎ ݊ ≥ 6
ሺ8. 10ିଵ+ ሺ10ିହ− 1ሻ. 10ସ+ 8877ሻ൬9. 10ିଵ+8
9 ሺ10ିଶ− 1ሻ. 10 + 7൰= 88. 10ଶିଶ+ ሺ10ି− 1ሻ. 10ାଶ+ 888. 10ିଵ+7
9 ሺ10ିଷ− 1ሻ. 10ସ+ 9899
6 898 877 988 887 888 887 779 899 ok
7 8 998 877 9 888 887 88 988 877 779 899 ok
8 89 998 877 98 888 887 8 899 888 777 779 899 ok
9 899 998 877 988 888 887 889 998 887 777 779 899 ok
10 8 999 998 877 9 888 888 887 88 999 988 877 777 779 899 ok
11 89 999 998 877 98 888 888 887 8 899 999 888 777 777 779 899 ok
12 899 999 998 877 988 888 888 887 889 999 998 887 777 777 779 899 ok
13 8 999 999 998 877 9 888 888 888 887 88 999 999 988 877 777 777 779 899 ok
14 89 999 999 998 877 98 888 888 888 887 8 899 999 999 888 777 777 777 779 899 ok
15 899 999 999 998 877 988 888 888 888 887 889 999 999 998 887 777 777 777 779 899 ok
16 8 999 999 999 998 877 9 888 888 888 888 887 88 999 999 999 988 877 777 777 777 779 899 ok
17 89 999 999 999 998 877 98 888 888 888 888 887 8 899 999 999 999 888 777 777 777 777 779 899 ok 18 899 999 999 999 998 877 988 888 888 888 888 887 889 999 999 999 998 887 777 777 777 777 779 899 ok 19 8 999 999 999 999 998 877 9 888 888 888 888 888 887 88 999 999 999 999 988 877 777 777 777 777 779 899 ok 20 89 999 999 999 999 998 877 98 888 888 888 888 888 887 8 899 999 999 999 999 888 777 777 777 777 777 779 899 ok 21 899 999 999 999 999 998 877 988 888 888 888 888 888 887 889 999 999 999 999 998 887 777 777 777 777 777 779 899 ok 22 8 999 999 999 999 999 998 877 9 888 888 888 888 888 888 887 88 999 999 999 999 999 988 877 777 777 777 777 777 779 899 ok 23 89 999 999 999 999 999 998 877 98 888 888 888 888 888 888 887 8 899 999 999 999 999 999 888 777 777 777 777 777 777 779 899 ok 24 899 999 999 999 999 999 998 877 988 888 888 888 888 888 888 887 889 999 999 999 999 999 998 887 777 777 777 777 777 777 779 899 ok 25 8 999 999 999 999 999 999 998 877 9 888 888 888 888 888 888 888 887 88 999 999 999 999 999 999 988 877 777 777 777 777 777 777 779 899 ok
8. Un huitième et un neuvième, pour lesquels nous n’avons pas trouvé de formulation mathématique simple, mais testés jusqu’à ݊ = 10 000
3 877 887 777 899 ok
4 7 877 9 887 77 879 899 ok
5 77 877 99 887 7 778 899 899 ok
6 877 877 999 887 877 777 799 899 ok
7 7 877 877 9 999 887 78 777 879 799 899 ok
8 77 877 877 99 999 887 7 787 778 899 799 899 ok
9 877 877 877 999 999 887 877 877 777 799 799 899 ok
10 7 877 877 877 9 999 999 887 78 778 777 879 799 799 899 ok
11 77 877 877 877 99 999 999 887 7 787 787 778 899 799 799 899 ok
12 877 877 877 877 999 999 999 887 877 877 877 777 799 799 799 899 ok
13 7 877 877 877 877 9 999 999 999 887 78 778 778 777 879 799 799 799 899 ok
14 77 877 877 877 877 99 999 999 999 887 7 787 787 787 778 899 799 799 799 899 ok
15 877 877 877 877 877 999 999 999 999 887 877 877 877 877 777 799 799 799 799 899 ok
16 7 877 877 877 877 877 9 999 999 999 999 887 78 778 778 778 777 879 799 799 799 799 899 ok
17 77 877 877 877 877 877 99 999 999 999 999 887 7 787 787 787 787 778 899 799 799 799 799 899 ok 18 877 877 877 877 877 877 999 999 999 999 999 887 877 877 877 877 877 777 799 799 799 799 799 899 ok 19 7 877 877 877 877 877 877 9 999 999 999 999 999 887 78 778 778 778 778 777 879 799 799 799 799 799 899 ok 20 77 877 877 877 877 877 877 99 999 999 999 999 999 887 7 787 787 787 787 787 778 899 799 799 799 799 799 899 ok 21 877 877 877 877 877 877 877 999 999 999 999 999 999 887 877 877 877 877 877 877 777 799 799 799 799 799 799 899 ok 22 7 877 877 877 877 877 877 877 9 999 999 999 999 999 999 887 78 778 778 778 778 778 777 879 799 799 799 799 799 799 899 ok 23 77 877 877 877 877 877 877 877 99 999 999 999 999 999 999 887 7 787 787 787 787 787 787 778 899 799 799 799 799 799 799 899 ok 24 877 877 877 877 877 877 877 877 999 999 999 999 999 999 999 887 877 877 877 877 877 877 877 777 799 799 799 799 799 799 799 899 ok 25 7 877 877 877 877 877 877 877 877 9 999 999 999 999 999 999 999 887 78 778 778 778 778 778 778 777 879 799 799 799 799 799 799 799 899 ok
3 887 877 777 899 ok
4 7 887 9 877 77 899 899 ok
5 87 887 99 877 8 777 889 899 ok
6 887 887 999 877 887 777 789 899 ok
7 7 887 887 9 999 877 78 877 899 789 899 ok
8 87 887 887 99 999 877 8 788 777 889 789 899 ok
9 887 887 887 999 999 877 887 887 777 789 789 899 ok
10 7 887 887 887 9 999 999 877 78 878 877 899 789 789 899 ok
11 87 887 887 887 99 999 999 877 8 788 788 777 889 789 789 899 ok
12 887 887 887 887 999 999 999 877 887 887 887 777 789 789 789 899 ok
13 7 887 887 887 887 9 999 999 999 877 78 878 878 877 899 789 789 789 899 ok
14 87 887 887 887 887 99 999 999 999 877 8 788 788 788 777 889 789 789 789 899 ok
15 887 887 887 887 887 999 999 999 999 877 887 887 887 887 777 789 789 789 789 899 ok
16 7 887 887 887 887 887 9 999 999 999 999 877 78 878 878 878 877 899 789 789 789 789 899 ok
17 87 887 887 887 887 887 99 999 999 999 999 877 8 788 788 788 788 777 889 789 789 789 789 899 ok 18 887 887 887 887 887 887 999 999 999 999 999 877 887 887 887 887 887 777 789 789 789 789 789 899 ok 19 7 887 887 887 887 887 887 9 999 999 999 999 999 877 78 878 878 878 878 877 899 789 789 789 789 789 899 ok 20 87 887 887 887 887 887 887 99 999 999 999 999 999 877 8 788 788 788 788 788 777 889 789 789 789 789 789 899 ok 21 887 887 887 887 887 887 887 999 999 999 999 999 999 877 887 887 887 887 887 887 777 789 789 789 789 789 789 899 ok 22 7 887 887 887 887 887 887 887 9 999 999 999 999 999 999 877 78 878 878 878 878 878 877 899 789 789 789 789 789 789 899 ok 23 87 887 887 887 887 887 887 887 99 999 999 999 999 999 999 877 8 788 788 788 788 788 788 777 889 789 789 789 789 789 789 899 ok 24 887 887 887 887 887 887 887 887 999 999 999 999 999 999 999 877 887 887 887 887 887 887 887 777 789 789 789 789 789 789 789 899 ok 25 7 887 887 887 887 887 887 887 887 9 999 999 999 999 999 999 999 877 78 878 878 878 878 878 878 877 899 789 789 789 789 789 789 789 899 ok
9. Récapitulatif
ܽ
ܾ
ܿ
8. 10ିଵ+7
9 ሺ10ିଵ− 1ሻ 8. 10ିଵ+ ሺ10ିଷ− 1ሻ. 10ଶ+ 87
78. 10
ଶିଶ+ ሺ10
ିସ− 1ሻ10
ାଶ+ 87.10
+ 8
9 ሺ10
ିଶ− 1ሻ10
ଶ+ 99 8877 8887 10ସିଵ− 1
10
ସ− 1 7 889. 10
ସ+ 7 788 10
ସሺିଶሻ− 1
10
ସ− 1 10
଼+ 77 879 899
89 ሺ10− 1ሻ. 10 + 7 ሺ10ିଷ− 1ሻ. 10ସ+ 8877
8
9 ሺ10
ିଷ− 1ሻ. 10
ାହ+ 7888. 10
ାଵ+ 7
9 ሺ10
ିଷ− 1ሻ. 10
ସ+ 9899
89 ሺ10ିସ− 1ሻ. 10ସ+ 7987 ሺ10ିଷ− 1ሻ. 10ଷ+ 7778
9 ሺ10
ିସ− 1ሻ. 10
ାସ+ 7788. 10
+ 7
9 ሺ10
ି− 1ሻ. 10
+ 978899
89 ሺ10ିଶ− 1ሻ. 10ଶ+ 77 ሺ10ି− 1ሻ. 10+ 898887
8
9 ሺ10
ି− 1ሻ. 10
ା+ 798998. 10
+ 7
9 ሺ10
ି− 1ሻ. 10
+ 8979899
78. 10ିଶ+79 ሺ10ିଶ− 1ሻ 98. 10ିଶ+ ሺ10ିସ− 1ሻ. 10ଶ+ 87 7798. 10ଶିସ+ ሺ10ି− 1ሻ. 10ାଶ+ 889.10ିଵ+8
9 ሺ10ିଷ− 1ሻ. 10ଶ+ 99 8. 10ିଵ+ ሺ10ିହ− 1ሻ. 10ସ+ 8877 9. 10ିଵ+8
9 ሺ10ିଶ− 1ሻ. 10 + 7 88. 10ଶିଶ+ ሺ10ି− 1ሻ. 10ାଶ+ 888. 10ିଵ+7
9 ሺ10ିଷ− 1ሻ. 10ସ+ 9899
