Exercises

1.
Find all the primes between 100 and 120. BITMAPSETAnswer0.2214in0.205in0ina1

2.
Find two positive integers between 1000 and 1100 whose greatest common divisor is 23.BITMAPSETAnswer0.2214in0.205in0ina2

1.
Compare the prime factorizations of 19!, 20!, 21!, !, 23!, 24!, and 25! with the number of zeros at the end of the evaluated factorials. Use these results to predict how many zeros will appear at the end of 100!. Check your prediction by direct evaluation. Revise your method of prediction, if necessary, and predict the number of zeros for each of 125!, 200!, 500!, 625!, and 1000!. BITMAPSETAnswer0.2214in0.205in0ina3

2.
Evaluate numerically the power

$\displaystyle \left(\vphantom{ 1+\frac{1}{n}}\right.$1 + $\displaystyle {\frac{{1}}{{n}}}$$\displaystyle \left.\vphantom{ 1+\frac{1}{n}}\right)^{{n}}_{}$

for n = 2, 4, 8, 16, 32, 64, 128 and 256. What well-known number is starting to emerge?BITMAPSETAnswer0.2214in0.205in0ina4

3.
Experiment with numbers to test the potential identities

a$\displaystyle \left(\vphantom{ b\vee c}\right.$bc$\displaystyle \left.\vphantom{ b\vee c}\right)$ = $\displaystyle \left(\vphantom{ a\wedge b}\right.$ab$\displaystyle \left.\vphantom{ a\wedge b}\right)$$\displaystyle \left(\vphantom{ a\wedge
c}\right.$ac$\displaystyle \left.\vphantom{ a\wedge
c}\right)$

and BITMAPSETAnswer0.2214in0.205in0ina5

a$\displaystyle \left(\vphantom{ b\wedge c}\right.$bc$\displaystyle \left.\vphantom{ b\wedge c}\right)$ = $\displaystyle \left(\vphantom{ a\vee b}\right.$ab$\displaystyle \left.\vphantom{ a\vee b}\right)$$\displaystyle \left(\vphantom{ a\vee
c}\right.$ac$\displaystyle \left.\vphantom{ a\vee
c}\right)$

4.
Test the potential identity

A$\displaystyle \left(\vphantom{ B\cup C}\right.$BC$\displaystyle \left.\vphantom{ B\cup C}\right)$ = $\displaystyle \left(\vphantom{ A\cap B}\right.$AB$\displaystyle \left.\vphantom{ A\cap B}\right)$$\displaystyle \left(\vphantom{ A\cap
C}\right.$AC$\displaystyle \left.\vphantom{ A\cap
C}\right)$

using the sets A = $\left\{\vphantom{ 1,3,5,7,9}\right.$1, 3, 5, 7, 9$\left.\vphantom{ 1,3,5,7,9}\right\}$, B = $\left\{\vphantom{ 1,4,9,16}\right.$1, 4, 9, 16$\left.\vphantom{ 1,4,9,16}\right\}$, and C = $\left\{\vphantom{ 2,3,5,7,11}\right.$2, 3, 5, 7, 11$\left.\vphantom{ 2,3,5,7,11}\right\}$.BITMAPSETAnswer0.2214in0.205in0ina6

5.
The mass of a block of aluminum is .2 $\unit$lb and the density is 168${\frac{{\unit{lb}}}{{\unit{ft}^{3}}}}$. What is its volume?BITMAPSETAnswer0.2214in0.205in0inVolume

6.
If a toy rocket shoots vertically upward with an initial velocity of 80 $\unit$m/$\unit$s, at t seconds after the rocket takes off, until it returns to the ground, it is at the height 80t - 16t2$\unit$m. Find the time it takes for the rocket to return to the ground. When does it reach its highest point?BITMAPSETAnswer0.2214in0.205in0inRocket