18.) Give reasons for each of the following steps (for example: place value, associative...):
35 x 100 = (3 x 10 + 5)100
= (3 x 10 + 5)10^2
= (3 x 10)10^2 + 5 x 10^2
= 3(10 x 10^2) + 5 x 10^2
= 3 x 10^3 + 5 x 10^2
= 3 x 10^3 + 5 x 10^2 + 0 x 10 + 0 x 1
= 3500
15.) a. Prove that –x – y = -y – x; for all integers x and y.
b. Does part (a) imply that subtraction is commutative? Explain.
18.) Find all integers x, if there are any, such that the following are true:
a. -|x| = 2
b. -|x| is negative.
c. -|x| is positive.
d. –x – 1 is positive.
e. –x – 1 is negative.
2.) Kahlil said that using the equation (a + b)^2 = a^2 + 2ab + b^2, he can find a similar equation for (a – b)^2. Examine his argument. If it is correct, supply any missing steps or justifications; if it is incorrect, point out why.
(a – b)^2 = [a + (-b)]^2
= a^2 + 2a(-b) + (-b)^2
= a^2 – 2ab + b^2
3.) Without using a calculator, test each of the following numbers for divisibility by 2, 3, 4, 5, 6, 8, 9, 10, and 11:
a. 4,201,012
b. 1001
c. 10,001
8.) Devise a test for divisibility by each of the following numbers:
a. 16
b. 25
2.) a. Jim uses his calculator to see if a number n having eight or fewer digits is divisible by a number d. He finds that n d has a display of 32. Does d|n? Why?
b. If n d gives a display of 16.8, does d|n? Why?
4.) Can you find three consecutive natural numbers none of which is divisible by 3? Explain your answer.
5.) Find the prime factorizations of the following:
a. 1001
b. 1001^2
c. 999^10
d. 111^10 - 111^9
6.) Suppose the 435 members of the House of Representatives are placed on committees consisting of more than 2 members but fewer than 30 members. Each committee is to have an equal number of members and each member is to be on only one committee.
a. What size committees are possible?
b. How many committees are there of each size?
2.) Explain why the product of any four consecutive integers is divisible by 24.
6.) A movie rental store gives a free popcorn to every fourth customer and a free movie rental to every sixth customer. Use the number-line method to find which customer was the first to win both prizes.
11.) The principal of Valley Elementary School wants to divide each of the three fourth-grade classes into small same-size groups with at least 2 students each. If the classes have 18, 24, and 36 students, respectively, what size groups are possible?