Needing help with problem 14 Prove that 12 13n 1 for every

Needing help with problem 14

Prove that 12 |(13^n - 1) for every Positive integer n. Express 234 as a product. For the following pairs m, n of integers, illustrate the Division Algorithm by finding integers q and r such that m = nq + r, where 0 lessthanorequalto r

Solution

14.a.. use the euclidean algorithm to determine gcd(70,42) .

sol: gcd(70,42)

70=1*42=28

42=1*28+14

28=2*14+0

gcd(70,42)=14

14.b. express gcd(70,42) as a linear combination of 70 and 42

sol: A=bq+r, where 0 < r < b

70=42*1+28

42=28*1+14

14=14*1+0

HCF=(70,42) = 14

now 14=42-28*1

14 = 42-(70-42*1)

14=42-70*1+42*1

14 = 42x+70y