Download Problems about NUMBERS The Beauty of Mathematics 1.

Problems about NUMBERS
The Beauty of Mathematics
1.- Suppose you are able to take a large piece of paper of ordinary thickness and fold it half 50 times. What would the height of the folded paper
be? About a yard? Taller than Mount Everest?
2.- The Earth has 6.200 million people and almost no one lives 100 years.
Suppose this longevity fact remains true. How do you know that some year
soon, more than 50 million people will die?
3.- By experimenting with numerous examples in search of a pattern,
determine a simple formula for the sum of the squares of two consecutive
Fibonacci numbers.
4.- Consider the following sequence of natural numbers: 1111, 11111,
111111, 1111111, 11111111, . . . Are all these numbers prime? If not, can you
describe infinitely many of these numbers that are definitely not prime?
5.- Suppose a certain number when divided by 13 yields a remainder of
7. If we add 22 to our original number, what is the remainder when this new
number is divided by 13?
6.- Find a run of six consecutive natural numbers, none of which is a prime
number. (Hint: Prove that you can start with (1 × 2 × 3 × 4 × 5 × 6 × 7) + 2.)
7.- A friend with lousy handwriting writes down a UPC. Unfortunately,
you can’t tell his 4’s from his 9’s or his 1’s from his 7’s. If the code looks like
9 0 3 0 6 8 8 2 3 5 1 7, is there any way to deal with the ambiguity? If so,
what is the actual UPC? If it is not possible to determine the correct UPC,
explain why.
8.- Suppose you wish to devise an RSA coding scheme for yourself. You
select p = 3 and q = 5. Compute m, and then find possible values for e and
d.
p
9.- Show that 2/3 is irrational.
10.- Express 6/7 in its decimal expansion and express 43.12121212. . . as
a fraction.