Question

can you please help me with these

Use u U 6. Use the Chinese remainder theorem to find all of the solutions to r? +1 = 0, modulo 1313. 7. What are the last two digits of 31000 ? 8. Find a positive integer x such that the last three digits of 77* are 007

Answer 1

solution::

Solution Gfiven that Finite tacto 1313 i 13 2 101 modulo 313 22 mod 1313 Andng The Soluton obove equivalent Soution = -I mod 13 to thi 2 and -1 mod 101 Y = 2 while asume Foa a mod 13 Y = -1 mod 10 Theorem ue have thi reye Remainda 13 x 101 13X 10 101 NI 13 13X1O 13 13 N2 =

Now line to Conquenus mod 10 - mod 3 Soluing we get 4, y 0 101X 4 mad 13 Imod 3 13*10 mod 10 = i mod 101 y= (X101x y )x13 x0 mod 1313 = -1314 mod 1313 -Imod 1313 131a med 1313 = 131 a mod 1313 48a mod 1313 Thus e get Tnemoind Solution 48a, 48 Veoifcation mod 1313 313

Solution Anumben N endi with 007 N= mod 1000 we have mod 1000 - 1 mod 1000 No Th 0nalen d mod 1000 is ao med C1000) iao divides n-1 want to Knoo ohen we 1 4 thu 0nde 7 mod a0 Noco -| mod O 4 hence