Integers and Divisibility

Counting (natural) numbers 1, 2, 3, …

Whole numbers 0, 1, 2, 3, …

Integers … -3, -2, -1, 0, 1, 2, 3, …

Absolute value - the distance between a number and zero

|4| =

| 2| =

4

2

Models for integer addition

Number Line Model

2 + –3 = 1

1 + 2 = 3

Charged field (chip) model for Addition

2 + 3 =

1 + 2 =

5

1

3 + 3 = 0

--

---

-+-

---

+++

Two numbers are additive inverses if their sum is zero.

Charged field (chip) model for Subtraction

2 ( 1) =

1 ( 2) =

1

3

--

+

“Take away”Start with 2 negativesTake away 1 negative1 negative left

Start with 1 positiveTake away 2 negatives3 positives left-

-

3

--

2 1 =

Start with 2 negativesTake away 1 positive3 negatives left

Number Line model for subtraction

2 1 = 3

-2-1

Negative numbers “face left” Positive numbers “face right”

Subtraction “walk backward”Addition “walk forward”

2 + ( 1) = 3

direction to face

how to walk

“Walk the line”

Relating subtraction to addition:a – b = a + (– b)

Number Line model for subtraction

2 ( 3) =

Negative numbers Positive numbers

Subtraction (backward)Addition (forward)

-2-(-3)

1

2 + (3) = 1

direction to face how to walk

Integer Multiplication

Charged Field (Chip) Model

3( 2) =

3(2) =

Add 3groups

of twonegatives

Take away3 groups

of twopositives

- -- -- -

6

+ +- - + +

- -

+ +- -

6

Number Line model for multiplication

3( 1) =

3( 1) =

3-1

Negative numbers Positive numbers

-(-1)

3

-1-1

3 groups

arrows

Of -1 arrows

Reverse3 groups

Of -1 arrows

-1 -(-1)-(-1)

Integer Multiplication – An investigation of patterns

3 · 3 =3 · 2 =3 · 1 =3 · 0 =

3 · 1 =3 · 2 =

3 · 3 =

9630369

3 · 3 =2 · 3 =1 · 3 =0 · 3 =1 · 3 =2 · 3 =

3 · 3 =

963

0369

Positive · Positive = PositivePositive · Negative = Negative

Positive · Negative = NegativeNegative · Negative = Positive

3

3

3

3

3

3

+3

+3

+3

+3

+3

+3

Same Signs – Positive answerDifferent Signs – Negative answer

Multiplication and Division

a · b = c means c b = a

Example: 3 · 4 = 12 means 12 4 = 3

Integer Division

pos · pos = pos so pos pos = pospos · neg = neg so neg neg = posneg · pos = neg so neg pos = negneg · neg = pos so pos neg = neg

Sign rules for division are identical to multiplication

Using the Difference of Squares formula to multiply

(a + b)(a – b) = a2 – b2

Multiply 42 · 38=(40 + 2)(40 – 2)= 402 – 22

= 1600 – 4= 1596

Multiply 107 · 93=(100 + 7)(100 – 7)= 1002 – 72

= 10000 – 49= 9951

Divisibility

If a and b are integers, then b divides a if there is an integer c such that a = b · c

Does 3 | 12

Does 6 | 12

Does 24 | 12

Yes

Yes

No

Because 12 = 3 · 4

Because 12 = 6 · 2

Because 12 = 24 · integer

Why?

Divisibility tests

234567891011

Even number (ends in 0, 2, 4, 6, 8)Sum of digits is divisible by 3Last two digits divisible by 4Ends in 0 or 5Divisible by both 2 and 3Cross out, double, subtractLast three digits divisible by 8Sum of digits divisible by 9Ends in 0Difference of alternate digits (ocean waves)

Test 5182 for divisibility by

2

3

4

5

6

Yes

No

No

No

No

Why?

5182 is even

5 + 1 + 8 + 2 = 16, and 3 | 18

4 | 82

5182 does not end in 0 or 5

Not divisible by both 2 and 3

Test 5182 for divisibility by

7

8

9

10

11

No

No

No

No

No

Why?

7 | 43

8 | 182

5 +1 + 8 + 2 = 16 and 9 | 16

5182 does not end in 0

11 | 10

5182- 4

514-843

5 1 8 2

13

3

13 – 3 = 10

Test 3,885,840 for divisibility by

2

3

4

5

6

Yes

Yes

Yes

Yes

Yes

Why?

3,885,840 is even

3+8+8+5+8+4+0=36, and 3 | 36

4 | 40

3,885,840 ends in 0

Divisible by both 2 and 3

Test 3,885,840 for divisibility by

7

8

9

10

11

Yes

Yes

Yes

Yes

No

Why?

7 | 21

8 | 840

9 | 36

Ends in 0

11 | 2

3,885,840- 0

388,584-8

38,850-0

3,885-10378

-1621

3 8 8 5 8 4 0

19

17 19 – 17 = 2