Question

how does 2 = 26 *5 -32 * 4 becomes 26 * 5 + 32 * (-6) = 2

Answer 1

We have,

2=26*5-32*4

That means 2 divides 26*5-32*4

That is 26*5-32*4 is congruent to 0(mod 2)

That is 26*5-32*4 0(mod 2)

26*5 32*4(mod 2) [since a-b 0(mod k)    a b(mod k) ]

Now we have,

32 (-32) (mod 2)

and 4 (-6) (mod 2) [since a b (mod k) and c d (mod k) ac bd (mod k) ]   

32*4 -32*(-6) (mod 2)

Thus we have

26*5 32*4 (mod 2) and 32*4 -32*(-6) (mod 2)

26*5 -32*(-6) (mod 2) [since a b (mod k) and bd (mod k) a d (mod k) ]

26*5 + 32*(-6) 0 (mod 2)

26*5 - 32*4 26*5 +32*(-6) (mod 2)

0 0 (mod 2)

  26*5 -32*4 = 2 = 26*5 + 32*(-6)