Show that -3 is a multiplicative inverse of 17 modulo 26.
-3 = 23 (mod 26)
17*23 = 26*15 + 1
To show that -3 is a multiplicative inverse of 17 modulo 26, we need to verify that their product is congruent to 1 modulo 26.
First, let's calculate the product of -3 and 17:
(-3) * 17 = -51
Next, we need to find the remainder when -51 is divided by 26. Since we are working with modulo 26, we only need to consider the remainder. The remainder when -51 is divided by 26 is 25.
So, (-3) * 17 ≡ 25 (mod 26)
Now, we need to check if 25 is congruent to 1 modulo 26. To do this, we can subtract multiples of 26 from 25 until we obtain a number that is congruent to 1.
If we subtract 26 from 25, we get -1. Since -1 is not congruent to 1 modulo 26, we need to continue subtracting.
If we subtract another 26 from -1, we get -27. Adding 26 to -27 gives us -1. Since -1 is also not congruent to 1 modulo 26, we need to continue.
Finally, if we subtract 3 more multiples of 26 from -1, we obtain 1. Therefore, we can conclude that 25 is congruent to 1 modulo 26.
Since (-3) * 17 ≡ 25 ≡ 1 (mod 26), we have shown that -3 is a multiplicative inverse of 17 modulo 26.