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Sagot :
The remainder will be 33, 9, 7, and 14 for options a, b, c, and d respectively.
The Remainder is the extra part of a number that restricts the number to be divided completely.
a)
(19² mod 41) mod 9
Let us determine 361 mod 41
a=361 = 328+33= 8*41+33 =8d+33
The remainder is the constant in the final expression: 361 mod 41=33
33 mod 9 = 6 (since a=33= 27+6=3*9+6=3d+6)
b)
(32³ mod 13)² mod 11
Let us determine 32 mod 13
a=32=26+6=2*13+6+2d+6
The remainder is the constant in the final expression: 32 mod 13=6
[tex](6^{3} mod13)^{2} mod 11\\(216mod13)^{2} mod 11[/tex]
Let us determine 216 mod 13
a= 216= 208+8=16*13+8 = 16d+8
The remainder is the constant in the final expression: 216 mod 13=8
[tex]8^{2} mod 11\\64 mod 11[/tex] = 9 (since, 64= 55+9= 11*5+6= 11d+6)
c)
(7³ mod 23)² mod 31
[tex](343mod23)^{2} mod 31\\[/tex]
Let us determine 343 mod 23
a=343=322+21=14*23+21=14d+21
The remainder is the constant in the final expression: 343 mod 23= 21
[tex]21^{2} mod 31 = 441 mod 31=7[/tex] (since a = 441=434+7=14*31+7=14d+7)
d)
(21² mod 15)³ mod 22
[tex](441mod15)^{3} mod 22[/tex]
Let us determine 441 mod 15
a=441 = 435+6= 29*15+6= 29d+6
The remainder is the constant in the final expression: 441 mod 15= 6
[tex]6^{2} mod 22=36mod22= 14[/tex] (since a = 36= 22+14= 1*22+14= d+14)
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