MathDB

p1. Find all prime factors of

8051

.
p2. Simplify

[\log_{xyz}(x^z)][1 + \log_x y + \log_x z],

where

x = 628

y = 233

z = 340

.
p3. In prokaryotes, translation of mRNA messages into proteins is most often initiated at start codons on the mRNA having the sequence AUG. Assume that the mRNA is single-stranded and consists of a sequence of bases, each described by a single letter A,C,U, or G. Consider the set of all pieces of bacterial mRNA six bases in length. How many such mRNA sequences have either no A’s or no U’s?
p4. What is the smallest positive

n

so that

17^n + n

is divisible by

29

?
p5. The legs of the right triangle shown below have length

a = 255

and

b = 32

. Find the area of the smaller rectangle (the one labeled

R

). https://cdn.artofproblemsolving.com/attachments/c/d/566f2ce631187684622dfb43f36c7e759e2f34.png
p6. A

3

dimensional cube contains ”cubes” of smaller dimensions, ie: faces (

2

-cubes),edges (

1

-cubes), and vertices (

0

-cubes). How many 3-cubes are in a

5

-cube?
PS. You had better use hide for answers.

Problem Statement