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SLIDE 1
What are the rules for divisibility of a number by 3, 4 and 9?
SLIDE 2
A number is divisible by 3 if the digit sum of the number is divisible by 3.
A number is divisible by 4 if the last two digits of the number are divisible by 4.
A number is divisible by 9 if the digit sum of the number is divisible by 9.
SLIDE 3
The four-digit number is divisible by 3 and 2. What is the value of X?
SLIDE 4
The four-digit number is divisible by 3 and 2. What is the value of X?
Solution
Divisible by 3 Divisible by 2 Solution
0
no ? no
1
? no no
2
no ? no
3
no no no
4
? ? ?
5
no no no
6
no ? no
7
? no no
8
no ? no
9
no no no
SLIDE 5
Positive integers a and b are multiples of 3 where . Which of the following must also be a multiple of 3? Give a counterexample for each case where not.
A
B
C
D
E
SLIDE 6
a and b are multiples of 3 where . Which of the following must also be a multiple of 3?
A no,
B no,
C ?
D no,
E no,
SLIDE 7
Positive integers a and b are multiples of 3.
Prove that is divisible by 3.
SLIDE 8
Positive integers a and b are multiples of 3
To prove that is divisible by 3
Proof
As a and b are both divisible by 3 then there exist integers m and n such that . Then,
Hence must be a multiple of 3, and so divisible by 3
SLIDE 9
Proof by contradiction
Positive integers a and b are multiples of 3.
To prove that cannot be divisible by 3.
Show that the assumption that leads to an impossible statement such as is an integer (a whole number). You may assume that is divisible by 3.
SLIDE 10
Positive integers a and b are multiples of 3.
Prove that cannot be divisible by 3.
Proof by contradiction
As a and b are both divisible by 3 then there exist integers m and n such that .
We have just shown that is divisible by 3. So,
where N is an integer, because is divisible by 3.
Suppose that is divisible by 3.
Then is an integer equal to , which is not an integer. This is impossible (a contradiction) because a number cannot be both an integer and not an integer.
Hence, cannot be divisible by 3.
SLIDE 11
a is an integer and b is a non-integer fraction. Which of the following cannot be an integer? If it can, give an example.
A
B
C
D
SLIDE 12
a is an integer and b is a non-integer fraction. Which of the following cannot be an integer? If it can, give an example.
A
B
C
D must be a non-integer fraction if b is a non-integer fraction and a is an integer.
SLIDE 13
If you add an irrational number to an integer is the result an irrational number, a rational number, or an integer?
SLIDE 14
If you add an irrational number to an integer is the result an irrational number, a rational number, or an integer?
Solution
Recall that an irrational number is something like the square root of a prime number and has an infinite non-repeating decimal expansion.
Adding an integer to an irrational number cannot result in anything other than an irrational number.
If you add a finite or infinitely repeating decimal expansion to an infinitely non-repeating expansion, you still have an infinitely non-repeating expansion.
SLIDE 15
A series of consecutive integers divisible by 3 has a sum that is a positive even number. The smallest number in the series is . What is the least possible number of integers in the series?
SLIDE 16
A series of consecutive integers divisible by 3 has a sum that is a positive even number. The smallest number in the series is . What is the least possible number of integers in the series?
Solution
We start with and check the sum as we go along, stopping at the first positive even sum.
Answer 8
SLIDE 17
n is a positive even integer. How many positive odd integers are less than n?
SLIDE 18
n is a positive even integer. How many positive odd integers are less than n?
Solution
Try something and look for the pattern. For example, suppose .
Then the number of positive odd integers is . If we add one more even integer to the sequence
we add both an odd and an even integer and the number of positive odd integers is . Hence, the solution is .
SLIDE 19
z is 6 more than y, and y is 2 less than x. If , what is z?
SLIDE 20
z is 6 more than y, and y is 2 less than x. If , what is z?
Solution
SLIDE 21
Let a be a two-digit square number and b be a two-digit cube number. If what is the largest possible two-digit value of N?
SLIDE 22
Let a be a two-digit square number and b be a two-digit cube number. If what is the largest possible two-digit value of N?
Solution
We begin by listing all the two-digit and cube numbers.
Two-digit square numbers: 16, 25, 49, 64, 81
Two-digit cube numbers: 27, 64
We list these in an array (a rectangular arrangement), putting the value of into the table.
Square number
16 25 49 64 81
Cube 27 5 23 71 101 135
number 64 -32 -14 34 64 98
From the table, the solution is 98, when the square number is 81 and the cube number is 64. We have
SLIDE 23
Starting with input of a positive integer x which of the following statements must be true?
I.
II. z is even
III. t is a number divisible by 3
SLIDE 24
Starting with input of a positive integer x which of the following statements must be true?
I.
II. z is even
III. t is a number divisible by 3
We annotate the flow diagram, which shows that statements II and III must be true. Note, if x is odd then is an odd number less an odd number, so must be even.
If we input , we obtain so statement I is not necessarily true. Answers II and III must be true (only).
SLIDE 25
The number line above has been divided into equal parts. What is the value of ?
SLIDE 26
The number line above has been divided into equal parts. What is the value of ?
Solution
The line has been divided into sevenths (seven equal parts).
SLIDE 27
Number puzzle on a triangle
The lengths of the sides of a triangle are a, b and c
Explain why each of the following is impossible.
? ?
? ?
SLIDE 28
Number puzzle on a triangle
The lengths of the sides of a triangle are a, b and c
Explain why each of the following is impossible.
? ?
? ?
Solution
? Suppose , then the supposed third side of the triangle is exactly the same length as the sum of the other two sides. A triangle cannot be drawn because it collapses into a straight line. For ?, rearranges to , so by the first argument it too must define a line and not a triangle. Hence in any triangle the length of the third side, c, must be such that
where are the lengths of the other two sides.
For both ? and ?, .
SLIDE 29
x and y are two positive integers such that and .
What is the value of ?
SLIDE 30
x and y are two positive integers such that and .
What is the value of ?
Solution
is the rule for the addition of fractions. Hence,
.
Then, .
SLIDE 31
Number puzzles on median and mean
When five integers are added together their sum is a number between 61 and 65. Which of the following could be the arithmetic mean of the numbers?
A 12
B 12.5
C 12.7
D 12.8
E 12.9
SLIDE 32
When five integers are added together their sum is a number between 61 and 65.
The mean is the sum divided by the total number. We multiply each of the options by 5, and check whether the product is a number between 61 and 65
A 12
B 12.5
C 12.7
D 12.8
E 12.9
The only possibility is D.
SLIDE 33
Four different positive odd numbers have a mean of 7. The median is larger than the mean. What are the four numbers?
SLIDE 34
Four different positive odd numbers have a mean of 7. The median is larger than the mean. What are the four numbers?
Solution
Imagine the numbers in rank order. For an even set of numbers, the median is the average of the two middle numbers. Since the median greater than the mean, which is 7, the third number in the list must be at least 9. Try 9 in that position and suppose the fourth number is 11.
Since the mean is 7, the total is . Since , the sum of the two smaller numbers must be . There are two possibilities, 1 and 7, or 3 and 5. If the second number in rank order is a 5, then the average of the two middle numbers, the median, is 7, which is not greater than 7. So, the first number must be 1 and the second 7.
No other solution is possible. For example, try 11 in the third place, or 13 in the fourth – they do not work. So,
the numbers in rank order must be 1, 7, 9 and 11. The average is 7 and the median is 8.
SLIDE 35
The arithmetic mean of 7 positive integers is a number greater than 7 and less than 12. The largest number in the list is 17. When this number is taken out of the set of 7 numbers, the mean is one less than before. What is the sum of the 7 numbers?
SLIDE 36
The arithmetic mean of 7 positive integers is a number greater than 7 and less than 12. The largest number in the list is 17. When this number is taken out of the set of 7 numbers, the mean is one less than before. What is the sum of the 7 numbers?
Solution
The mean must be one of the following numbers, 8, 9, 10 or 11.
7 numbers 6 numbers
mean total mean total difference
8 56 7 42 14
9 63 8 48 15
10 70 9 54 16
11 77 10 60 17
The difference between the two totals is equal the largest number removed. Thus, the mean of the seven numbers must be 11, and their sum 77.
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