Find n such that n ? 9 (mod 10) n ? 9 (mod 11) n ? 0 (mod 13)

1
Find n such that n ? 9 (mod 10) n ? 9 (mod
11) n ? 0 (mod 13)
Problem
solution

• Z1 11x13 Z210x13 Z310x11
  

Y1 ? Z1-1 (mod10) Y2 ? Z2-1 (mod11) Y3 ?
Z3-1 (mod13)
n ? 9xZ1xY1 9xZ2xY2 0xZ3xY3 (mod 10x11x13)
2
n ? 9 (mod 10) n ? 9 (mod 11)
n ? 0 (mod 13) n ?
Solution

• Z1 143 Z2 130

Y2 ?Z2-1 (mod 11) ? 130-1( mod11) ?
9-1 (mod11) ( since 130 ?9 (mod11)) ? 5
(mod11) (since 9x5 ?45 ?1(mod11))
Y1 ? Z1-1 (mod10) ? 143 -1(mod10) ?
7 (mod10) Since 143x7 ? 1001 ? 1 (mod10)
11 9x1 2 92x4 11 9-2x4 9-(11-9)x4 5x9
-4x11
n ? 9x143x7 9 x 130x5 (mod10x11x13) ? 559
(mod 1430)
3

• n ?0 (mod13) ? n13k
• n ? 9 (mod10) ? 13k ?9 (mod10)
• k ?91k ? 7x13k ? 7x9 ? 63 ?3 (mod10) ? k
  10l 3
• n ?9 (mod11) ? 13k ? 9 (mod11)
• 13(10l3) ?9 (mod11) ? 130l ? -30 (mod11)
• l ? 650l ? 5x130l ? 5x(-30) ?-150 ?4 (mod11) ? l
  11t4
• n 13k130l39 130(11t4)39 1430t 559