Take Home Exam 3 Solved

Question 1

Let p be a prime, x be a positive integer which is not divisible by p, and y be the smallest positive integer where xy 1 (mod p). Prove that y j (p 1).

Question 2

Show that 169 - (2n2 + 10n    7), 8n 2 Z+.

Question 3

Let a and b be integers and m and n be positive integers. Given a b (mod m) and a b (mod n) where gcd(m; n) = 1 prove that a b (mod m n).


Question 4

Use mathematical induction to prove that for all positive integers k and n,

n
n(n+1)(n+2)  (n+k)
jP


(k+1)
j(j + 1)(j + 2)   (j + k  1) =

=1


Question 5

Let H0 = 1, H1 = 3, H2 = 5, and de ne

Hn = 5Hn  1 + 5Hn  2 + 63Hn  3

for n    3. Show by strong induction that Hn    7n for all n    0.







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