Question

If x is congruent to 13 modulo 17 then 7x – 3 is congruent to which number modulo 17?

Answer 1

x ≡ 13 (mod 17)

Let p be the required number   ...(1)

7x – 3 ≡ p (mod 17)   ...(2)

From (1),

x – 13 = 17n for some integer M.

x – 13 is a multiple of 17.

x must be 30.

∴ 30 – 13 = 17

which is a multiple of 17.

From (2),

7 × 30 – 3 ≡ p (mod 17)

210 – 3 ≡ p (mod 17)

207 ≡ p (mod 17)

207 ≡ 3 (mod 17)

∴ P ≡ 3