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Properties of Numbers
SLIDE 1
Work out
? ?
? ?
? ?
? ?
? ?
SLIDE 2
? ?
? ?
? ?
? ?
? ?
SLIDE 3
? How can we tell whether a number can be divided by 2 just by looking at the number?
? Which of the following numbers are divisible by 2?
89, 191, 32, 80, 37, 1919, 2, 0, 1338766
SLIDE 4
? A number divisible by 2 is an even number. All the other numbers are odd numbers.
Odd and even numbers alternate. The even numbers end in
0, 2, 4, 6, or 8
A number is divisible by 2 if it ends in a 0, 2 ,4, 6 or 8.
? 89 ? odd, 191 ? odd, 32 ? even, 80 ? even, 37 ? odd, 1919 ? odd, 2 ? even, 0 ? even, 1338766 ? even.
Notes for tutors
The number 0 (zero) presents some difficulties, and one will even see some contradictory statements about whether it is odd or even. It is a special number, and whether it is even or not to some extent depends on what we define an even number to be. Divisibility is not the fundamental property here, but rather parity, which is the property of alternating sequences. We keep the alternating property of odd, even, odd, even, … going, and require zero to be an even number. Another way of looking at this is that a number is even if it can be written as where n is an integer. Here . The property of odd may also be more fundamental. A number is odd if it can be written as . Zero is not odd, so it is even.
Discussion of the alternating property of numbers, odd, even, odd, even … is possible with the student, and may awaken interest.
SLIDE 5
? How can we tell whether a number can be divided by 5 just by looking at the number?
? Which of the following numbers are divisible by 5?
65, 14, 125, 3, 50, 41, 152, 1010, 134565, 0, 1
SLIDE 6
? A number is divisible by 5 if it ends in a 0 or 5
? Which of the following numbers are divisible by 5?
65 ? yes, 14 ? no, 125 ? yes, 3 ? no, 50 ? yes, 41 ? no, 152 ? no, 1010 ? yes, 134565 ? yes, 0 ? yes, 1 ? no.
SLIDE 7
149 is a number.
In this number, 1, 4 and 9 are digits.
The first digit of the number 149 is 1.
The second digit of the number 149 is 4.
The third digit of the number 149 is 9.
? What is the first digit of the number 61?
? What is the second digit of the number 293?
? What is the last digit of the number 3758?
? What is the third digit of the number 653290?
SLIDE 8
? The first digit of the number 61 is 6.
? The second digit of the number 293 is 9.
? The last digit of the number 3758 is 8.
? The third digit of the number 653290 is 3.
SLIDE 9
Divide the following numbers by 4
100, 200, 300, 400, 500, 600, 700, 800, 900
What do you notice?
SLIDE 10
We see that all multiples of 100 are divisible by 4.
Whether a number larger than 100 is divisible by 4 or not depends on the last two digits of the number only.
SLIDE 11
? For the numbers 30 to 40 find the remainder when the number is divided by 4
? For the numbers 130 to 140 find the remainder when the last two digits is divided by 4.
Number Remainder
divided by 4 Number Remainder of last two digits divided by 4
30 130
31 131
32 132
33 133
34 134
35 135
36 136
37 137
38 138
39 139
40 140
SLIDE 12
Number Remainder
divided by 4 Number Remainder of last two digits divided by 4
30 2 130 2
31 3 131 3
32 0 132 0
33 1 133 1
34 2 134 2
35 3 135 3
36 0 136 0
37 1 137 1
38 2 138 2
39 3 139 3
40 0 140 0
SLIDE 13
Rule for divisibility by 4
A number is divisible by 4 if the number made by the last two digits is divisible by 4.
? Is 256 divisible by 4?
? Is 3722 divisible by 4?
? Is 1988 divisible by 4?
? Is 2394 divisible by 4?
SLIDE 14
? Is 256 divisible by 4?
? 256 is divisible by 4
? Is 3722 divisible by 4?
22 is not divisible by 4 ? 3721 is not divisible by 4
? Is 1988 divisible by 4?
? 1988 is divisible by 4
? Is 2394 divisible by 4?
94 is not divisible by 4 ? 2394 is not divisible by 4
SLIDE 15
To partition means to separate into two collections.
By assigning the following numbers to one of two collections, partition them into numbers divisible by 4 and numbers not divisible by 4
240 315 420 1680 1710 346
6888 2030 1555 2442 5672
6728 1200 561 2436 9120 8326
Divisible by 4 Not divisible by 4
SLIDE 16
240 315 420 1680 1710 346
6888 2030 1555 2442 5672
6728 1200 561 2436 9120 8326
Divisible by 4
240 420 1680
6888 5672 6728
1200 2436 9120 Not divisible by 4
315 1710 346
2030 1555 2442
561 8326
SLIDE 17
Long division by 2
This is read
2 goes into 5 ? 2 times remainder 1
2 goes into 17 ? 8 times remainder 1
2 goes into 13 ? 6 times remainder 1
In this example, the number 286 is called the quotient, 2 is the divisor and 573 is the dividend.
The final remainder is shown by the letter r
SLIDE 18
Complete the following
?
?
?
SLIDE 19
?
?
?
Notes for tutors
It is not necessary to show the zeros, and these are usually omitted.
If three examples are not sufficient for the student, then make up examples of your own.
SLIDE 20
Long division by 3
This is read
3 goes into 7 ? 2 times remainder 1
3 goes into 14 ? 4 times remainder 2
2 goes into 29 ? 7 times remainder 1
SLIDE 21
Complete the following
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?
?
SLIDE 22
?
?
?
SLIDE 23
The digit sum of a number is the sum (total) of all the digits in the number.
Example
The digit sum of the number 9235 is .
Exercise
Find the digit sum of the following numbers.
? 1357
? 24683
? 50354
? 89113
SLIDE 24
? 1357 digit sum =
? 24683 digit sum =
? 50354 digit sum =
? 89113 digit sum =
SLIDE 25
For each the numbers in the table
(a) Find the remainder when divided by 3
(b) Find the digit sum
(c) Find the remainder when the digit sum is divided by 3
Complete the table
Number Remainder
divided by 3 Digit sum Remainder of digit sum divided by 3
90 0 9 0
372
4572
35484
88
424
5917
65431
74
836
2525
47342
SLIDE 26
Number Remainder
divided by 3 Digit sum Remainder of digit sum divided by 3
90 0 9 0
372 0 12 0
4572 0 18 0
35484 0 24 0
88 1 16 1
424 1 10 1
5917 1 23 1
65431 1 19 1
74 2 11 2
836 2 17 2
2525 2 14 2
47342 2 20 2
SLIDE 27
Rule for divisibility by 3
A number is divisible by 3 if the digit sum of the number is divisible by 3
? Is 256 divisible by 3?
? Is 3723 divisible by 3?
? Is 1988 divisible by 3?
? Is 2394 divisible by 3?
SLIDE 28
Rule for divisibility by 3
A number is divisible by 3 if the digit sum of the number is divisible by 3
? Is 256 divisible by 3?
Digit sum =
13 is not divisible by 3 ? 256 is not divisible by 3
? Is 3723 divisible by 3?
Digit sum =
15 is divisible by 3 ? 3723 is divisible by 3
? Is 1988 divisible by 3?
Digit sum =
26 is not divisible by 3 ? 26 is not divisible by 3
? Is 2394 divisible by 3?
Digit sum =
18 is divisible by 3 ? 2394 is divisible by 3
SLIDE 29
A set is another name for a collection
Partition the following set into two sets
numbers divisible by 3 numbers not divisible by 3
813 437 555 3548 1722 753
6889 2031 5555 3881 6530
7588 1212 563 2436 9122 8973
Divisible by 3
Not divisible by 3
SLIDE 30
813 437 555 3548 1722 753
6889 2031 5555 3881 6530
7588 1212 563 2436 9122 8973
Divisible by 3
813 555 1722
753 2031 1212
2436 8973 Not divisible by 3
437 3548 6889
5555 3881 6530
7588 563 9122
SLIDE 31
Divisible by
Number 2 3 4 5 10
380 ? ? ? ? ?
4955
58122
65494
32211
94487
75230
SLIDE 32
Divisible by
Number 2 3 4 5 10
380 ? ? ? ? ?
4955 ? ? ? ? ?
58122 ? ? ? ? ?
65494 ? ? ? ? ?
32211 ? ? ? ? ?
94487 ? ? ? ? ?
75230 ? ? ? ? ?
SLIDE 33
The factors of 18 in ascending order are
Find the all the factors in ascending order of these numbers
? 20 ? 24
? 32 ? 25
? 36 ? 60
? 72 ? 84
SLIDE 34
? 20
? 24
? 32
? 25
? 36
? 60
? 72
? 84
SLIDE 35
SLIDE 36
SLIDE 37
SLIDE 38
SLIDE 39
SLIDE 40
SLIDE 41
SLIDE 42
A prime number is a number that can be divided only by itself and 1.
The first five prime numbers are
2 3 5 7 11
The numbers 0 and 1 are defined to be neither prime nor not prime.
The smallest prime number is 2, and it is the only even prime number.
To find prime numbers test whether the number can be divided by smaller prime numbers.
Example
Test whether 13 is not prime.
13 cannot be divided by 2
13 cannot be divided by 3
13 cannot be divided by 5
13 cannot be divided by 7
Therefore, 13 is a prime number
Notes for Tutors
When testing a number it is only necessary to use numbers up to the square root of the number. This is because when we divided one number by another we always get a pair of numbers. For example,
For every smaller factor there is a larger factor. So, if one has not found a factor by the time one reaches the square root of the number, then the number is prime.
This procedure (an algorithm) is known as the sieve of Eratosthenes.
Eratosthenes of Cyrene was a 3rd century BC Greek mathematician who was the first to calculate the circumference of the Earth, the tilt of the Earth’s axis and may have calculated the distance of the Earth from the Sun. He introduced the leap year and was the first to make a projective map of the Earth. He was chief librarian at Alexandria.
SLIDE 43
Test whether 14, 15, 16, 17, 18, 19 and 20 are prime numbers
SLIDE 44
A number that is not prime is a composite number.
14 composite
15 composite
16 composite
17 no divisors prime
18 composite
19 no divisors prime
20 composite
SLIDE 45
Find all prime numbers up to 100.
Count the number of prime numbers in each decade, that is from 0 to 10, from 11 to 20, and so on.
What do you observe about the number of prime numbers in each decade?
SLIDE 46
Decade Primes Number of primes
0 to 10 2, 3, 5, 7 4
11 to 20 11, 13, 17, 19 4
21 to 30 23, 29 2
31 to 40 31, 37 2
41 to 50 41, 43, 47 3
51 to 60 53, 59 2
61 to 70 61, 67 2
71 to 80 71, 73, 79 3
81 to 90 83, 89 2
91 to 100 97 1
Although the pattern is uneven, the number of prime numbers in each decade tends to go down. It starts at 4 in the first two decades and decreases to 2.
SLIDE 47
? Why does the number of prime numbers in each decade go down on average?
? Complete the two rows of this table
number construction
2 2 prime
7
prime
31
prime
221 prime
2321 prime
? Could there be a largest prime number?
SLIDE 48
? There are more and more prime numbers, so we have more and more ways of making a composite number. Thus, the number of primes goes down.
?
number construction
2 2 prime
7
prime
31
prime
221
prime
2311
prime
? Could there be a largest prime number?
There cannot be a largest prime number because the method above shows how to construct (make) a larger prime number from small prime numbers.
Notes for tutors
The method in constructing a prime number is used in the proof that there is no largest prime number. The construction is: list all the prime numbers in sequence up to a given prime. Multiply all these numbers together, and then add 1 to that product. The result is a prime number.
SLIDE 49
Any number that has a definite size is finite.
All counting numbers, 0, 1, 2, 3, 4, … are finite.
If something is not finite, then it is infinite.
The number of prime numbers is infinite.
SLIDE 50
A twin prime is any pair of prime numbers separated only by a single even number.
3 and 5 are twin primes.
5 and 7 are twin primes.
? The prime numbers up to 100 are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97
Find all the twin primes in this list.
? Could there be a largest pair of twin primes? Alternatively, is the number of twin primes infinite?
SLIDE 51
3 and 5 are twin primes.
5 and 7 are twin primes.
? The prime numbers up to 100 are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91 and 97
Find all the twin primes in this list.
3, 5 5, 7 11, 13 17, 19
29, 31 41, 43 59, 61 71, 73
? The statement that there are infinitely many twin primes is known as the twin prime conjecture.
To this day it is not known whether this conjecture is true or false.
At September 2018 the largest twin prime pair known has 388,342 decimal digits. There are 808,675,888,577,436 twin prime pairs below a million million million.
There is a large cash prize for the person who solves the problem either way.
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