1. n Express the number (12.34)5 in base 10 as m where m, n are positive integers such that m, n do not have common divisor larger than 1, that is, gcd(m, n) = 1.

Prove that 52N-1 is divisible by 8 for all NEN 5²-1 - 8 2(1) 1) Show for N=1 5203-1-N 8 52h-Y8 3) Apply K+1: 52 (H+1)_1 = 2) Assume K: · L = N 1=3 yes, divisible

Use the factorization A = PDP-1 to compute Ak, where k represents an arbitrary positive integer. 2-142 1³:1 = 01 AK = - 15 72 -4 19 49 30 -2 9 1 -4

Let x be a natural number with a binary representation where the difference between the number of 1 bits with an even index and the number of 1 bits with an odd index is a multiple of 3. Prove that x is a multiple of 3.

Define the fibonacci numbers by f(0) = 0; f(1) = 1; and then recursively for natural numbers n ≥ 2 by f(n) = f(n-1) + f(n-2). a) Write out the first 7 fibonacci numbers. I b) Make a conjecture about which numbers n will give us that f(n) will be an even number. c) Prove an exact characterization of these numbers using strong induction. (i.e. show that if your characterization is true up to some point, then it would be true just after that)

Expand (x+4) using Pascal's triangle. 1. What is the correct row of numbers to use from Pascal's triangle? A) 1, 3, 3, 1 B) 1, 4, 6, 4, 1 C) 1, 3, 6, 3, 1 D) 1, 2, 1 2. Which of the following shows the correct expression to expand the binomial? A) 1x'(4)* +2x²(4)² +1x*(4}° B) 1x°(4)° +3x'(4)* +3x*(4}* +1x°(4}³ C) 1x°(4)° +3x²(4)* +3x*(4)° + ix°{4}³ D) 1x'(4)° +(3x}(4)' +(3x)*(4)* +1x°(4)³ 3. Write the expansion in simplified form. A) x³ +36x² +144x+64 B) 1+12x+48x² +64x³ C) x +36x? + 48x +64 D) x' +12x? +48x+64

2- 3" – 1. 3an + 2, for all nonnegative integers n. Prove that an %3D Let ao 1, and let an+! %3D