By partitioning

10!=1\\\\times 2\\\\times 3\\\\times 4\\\\times 5\\\\times 6\\\\times 7\\\\times 8\\\\times 9\\\\times 10

as\

(1\\\\times 3\\\\times 5)(7\\\\times 9)(2\\\\times 4\\\\times 6\\\\times 8\\\\times 10)

\ removing -1 and 2 from appropriate factors, and then changing negative factors

-k

\ back to positive

11-k

, show that\

10!-=10!(-1)^(3)2^(5),(mod11)

\ and hence that

((2)/(11))=-1

partitioning 10!=12345678910 as (135)(79)(246810) removing -1 and 2 from appropriate factors, and then changing negative factors k back to positive 11k, show that 10!10!(1)325(mod11) and hence that (112)=1. partitioning 10!=12345678910 as (135)(79)(246810) removing -1 and 2 from appropriate factors, and then changing negative factors k back to positive 11k, show that 10!10!(1)325(mod11) and hence that (112)=1