Your Algebra Homework Can Now Be Easier Than Ever!

PRIMES AND DIVISIBILITY

PRIMES AND DIVISIBILITY: PRACTICE PROBLEMS

1) True or False? If false, then correct with a similar statement. (In this
problem, assume that variables stand for natural numbers.)

(a) If n is even, then n/2 is even.
(b) If n is even, then 4|2n.
(c) If n is even, then 4|n2.
(d) The general form for a multiple of three is 2a + 1.
(e) If 2|a and 5|b, then a2b is a multiple of 50.
(f) The sum of two odd numbers is never a multiple of three.
(g) If 3|a and 7|b, then 10|a + b.
(h) If 3|a and 9|b, then 3|a + b.
(i) The sum of any three consecutive numbers is divisible by three.
(j) The product of any three consecutive numbers is divisible by six.
(k) If c|ab, then c|a or c|b.
(l) If d|a, then d2|a3.
(m) The square of any odd number leaves remainder 1 when you divide it
by 8.
(n) The least common multiple of a and b always divides ab.

(2) Is 101 prime? Explain how to check this efficiently.

(3) (Venn diagrams and divisibility) For each of the following, use a Venn-type
diagram to efficiently represent the relationship between the following kinds
of number.

(a) Multiples of 2, Multiples of 3, Multiples of 6
(b) Multiples of 8, Multiples of 6, Multiples of 5
(c) Multiples of 4, Multiples of 16, Multiples of 64
(d) Multiples of 15, Multiples of 25
(What is a description of the intersection?)

(4) Any four digit palindrome ( like 1331 or 2772) is divisible by 11. Why? Is
it true for five- or six-digit palindromes?

(5) Give the prime factorization of each of the following numbers: 48, 100, 66,
504.

(6) Devise and justify a divisibility rule for the following numbers: 18, 24, 25
in base 10; 2 and 4 in base 5; 3 in base 12. Give some examples of easy
divisibility rules in other bases.

(7) Give an example of a number with exactly five divisors. Give an example
of a number with exactly 100 divisors.

(8) Find the smallest natural number that is divisible by all of the numbers up
to 10. How else could this question have been worded?

(9) What is the smallest composite number that is not divisible by 2, 3, 5, or
7?

(10) What is the ones digit of 7100?

(11) Find the gcd and lcm of the following pairs: (30, 75), (20, 40), (105, 132), (902, 908).
(You don’t have to simplify your answer .)

(12) The gcd of two numbers is 18. Their lcm is 630. What are the two numbers?

(13) * Suppose there is a row of one million pennies, all with the heads side
facing up. Now someone goes through and flips over every second coin
(that is, the ones numbered 2, 4, 6, and so on are flipped to the tails side).
Now someone goes through and flips over every third coin (so they turn
over 3, 6, 9, and so on: now 3 is tails up but 6 is turned back to heads
up). This process is continued for every divisor from two to a million. Will
the coin numbered 2008 be heads up or tails up at the end of the process?
Which coins are heads up?

(14) * Find five consecutive composite numbers over 1000. Give a method for
finding ten
consecutive composite numbers.

Prev Next

Start solving your Algebra Problems in next 5 minutes!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of March 28th you will pay only $39.99 instead of our regular price of $74.99 -- this is $35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from tutor.com for a 1$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:



Algebra Helper can solve problems in all the following areas:

  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

ORDER NOW!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath
Check out our demo!
 
"It really helped me with my homework.  I was stuck on some problems and your software walked me step by step through the process..."
C. Sievert, KY
 
 
Sofmath
19179 Blanco #105-234
San Antonio, TX 78258
Phone: (512) 788-5675
Fax: (512) 519-1805
 

Home   : :   Features   : :   Demo   : :   FAQ   : :   Order

Copyright © 2004-2024, Algebra-Answer.Com.  All rights reserved.