For a-b, reduce to the least non-negative residue

a. (88 · 95 · 36 · 703) mod 7

b. 4^ (83) mod 11

a. 6

b. 9

Step-by-step explanation:

a. The product modulo 7 can be found from the product of the individual numbers modulo 7:

(88·95·36·702) mod 7 = (88 mod 7) · (95 mod 7) · (36 mod 7) · (703 mod 7) mod 7

= (4·4·1·3) mod 7 = 48 mod 7 = 6

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b. Powers of 4 mod 11 repeat with period 5:

4 mod 11 = 4

4^2 mod 11 = 5

4^3 mod 11 = 9

4^4 mod 11 = 3

4^5 mod 11 = 1

So, 4^83 mod 11 = 4^3 mod 11 = 9