Solutions for Session 1, Part A

See solutions for Problems: A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | A10 | A11

 Problem A1 Here is the completed table:

Problem A2

 a. Answers will vary. Many people will notice that the table is symmetrical; if you look at any two points directly across the main diagonal (from top left to bottom right), you will see that the values are equal. Other patterns include the fact that the same number appears on each diagonal from bottom left to top right (one diagonal has nothing but 9s, for example), and the fact that each number appears in every row and column exactly once. b. The diagonal pattern exists because these are all numbers that add up to the same value in the real number system. Moving one unit up reduces the sum by 1, and moving one unit to the right increases the sum by 1, so moving along this type of diagonal (up and right, or down and left) does not change the sum of the two numbers.

 Problem A3 The number is 0. Zero plus any number in the system equals that number. For example, 7 + 0 and 0 + 7 both equal 7.

Problem A4

a.

For the number 4, we are looking for a number x so that 4 + x = 0 in the system. We want a value so that 0 will be the units digit of the sum. If the sum is 10, then 0 will be the units digit. Therefore, x is 6, and 4 + 6 = 0 in the system.

b.

Yes, every number in the table has an additive inverse. Here is a table:

Number

Inverse

 0 0 1 9 2 8 3 7 4 6 5 5 6 4 7 3 8 2 9 1

Note that 0 and 5 are their own inverses.

Problem A5

 a. Yes, this law holds. The main reason is that the law is true in real numbers, so it must also be true in this system (which is based directly on real number arithmetic). You can also see this from the table; (a + b) and (b + a) are opposite each other on the main diagonal. b. Yes, this law holds, since the same law is true in real numbers.

Problem A6

 a. Finding the answer to 7 - 3 is the same as finding a number x so that 3 + x = 7 in the system. To find this number, we can look at Row 3 in our table to see all the possible results we can get from 3 + x. In this case, the result in Column 4 gives 7, so 3 + 4 = 7, and 4 (the column value) is the solution to 7 - 3. As a more complicated example, let's find 2 - 9. This is the same as finding a number x so that 9 + x = 2 in the system. Looking in the row for 9, we want to find a result of 2. This happens in Column 3, so we know that 9 + 3 = 2 and that 3 is the solution to 2 - 9. b. Yes, it is possible to subtract any number from any other number in this system. This is true because each number occurs in every row and column exactly once, so we can always find a solution to a + x = b, no matter what numbers a and b are.

 Problem A7 Yes, definitely. When we add two numbers in the system, we always get a number in the system. Remember that we are looking at only the units digit of the solution, so 6 + 7 = 3, not 13.

 Problem A8 Here is the completed table:

Problem A9

 a. Answers will vary. For example, one pattern is that the first row and column are all zeros. Another is that the table is symmetrical about its main diagonal. Another is that some (but not all) rows have all 10 numbers. b. Answers will vary. Some patterns are pretty easy to explain -- every number multiplied by 0 is 0, so the row and column for 0 should be nothing but zeros. Some are much more difficult to explain, such as which rows will have all 10 numbers.

Problem A10

 a. Let's start with 3. We want to find a number m so that m • 3 = 3 • m = 3. Looking at the row and column for 3, the only number that works is m = 1. b. Yes, m = 1 works for every number in the table.

Problem A11

 a. We want to find a number f so that 3 • f = 1. Looking at the row for 3, the only number that works is f = 7. So 7 is the multiplicative inverse of 3. b. The inverse of 1 is 1. The inverse of 3 is 7. The inverse of 7 is 3. The inverse of 9 is 9. These are the only numbers that have inverses in this system. c. Numbers with inverses: 1, 3, 7, 9. Numbers without inverses: 0, 2, 4, 5, 6, 8. All the numbers with inverses are odd, while every even number has no inverse. The only exception to this pattern is 5; according to the multiplication table, any number multiplied by 5 will have a units digit of 0 or 5, so 1 is never a units digit. More explicitly, the numbers with inverses are relatively prime to 10 (they have no common factors, except 1, with 10). The numbers without inverses are not relatively prime to 10; they have common factors with 10 that are greater than 1.

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