Directions: Prove or disprove the following statements using counterexample.
Write a short
explanation of your step-by-step process for each test item.
1. All numbers that end in 1 are prime numbers.
Solution:
First, I have to list some numbers that ends in 1.
11, 21, 31, 41, 51, 61, 71, 81, 91, 101, …
Next, I will check if all the numbers in the list are prime numbers.
11 – prime
21 – not prime (divisible by 3 and 7 other than 1 and itself)
31 – prime
41 – prime
51 – not prime (divisible by 3 and 17 other than 1 and itself)
61 – prime
71 – prime
81 – not prime (divisible by 3, 9, and 27 other than 1 and itself)
91 – prime
101 – prime
Since the numbers, 21, 51, and 81 is not only divisible by 1 and itself. In this
example, I can say that not all numbers that end in 1 are prime numbers. So I call 21,
51, and 81 as counterexamples.
2. The sum of any two whole numbers is divisible by 2.
Solution:
First, I have to list any two whole numbers and get the sum.
1+1=2
1+3=4
3+4=7
9 + 3 = 12
11 + 8 = 19
Next, I will check if all the sum of any two whole numbers in the list is divisible by
2.
2/2=1
4/2=2
7 / 2 = 3.5
12 / 2 = 6
19 / 2 = 9.5
Since the quotient of 7 and 2 is 3.5 and 19 and 2 is 9.5, which is not exact, so I
can say that 7 and 19 is not divisible by 2. With this example, I have shown that not all
the sum of any two whole numbers is divisible by 2.
Activity 4.2
Directions: Use inductive and deductive reasoning in answering the problems below. Write a
short explanation of your step-by-step process for each test item.
1. 8, 11, 14, 17, 20, 23, 26, 29, …
Looking at the terms in the given sequence, I noticed that the succeeding terms are
obtained by adding 3. Hence, I can conclude that the next terms will also be the sum of the
previous number and 3. Since we are looking for the 6th, 7th, and 8th terms, then I add the 5th
term by 3 and I’ll get the 6th term and so on.
2. 105, 95, 85, 75, 65, 55, 45, 35, …
Looking at the terms in the given sequence, I noticed that the succeeding terms are
obtained by subtracting 10. Hence, I can conclude that the next terms will also be the difference
of the previous number and 10. Since we are looking for the 6th, 7th, and 8th terms, then I
subtract the 5th term by 10 and I’ll get the 6th term and so on.
3. 32, 3.2, 0.32, 0.032, 0.0032, 0.00032, 0.000032, …
Looking at the terms in the given sequence, I noticed that the succeeding terms are
obtained by dividing 10. Hence, I can conclude that the next terms will also be the quotient of
the previous number and 10. Since we are looking for the 5th, 6th, and 7th terms, then I divide the
4th term by 10 and I’ll get the 5th term and so on.
4. Show that when a number is multiplied by 6, then 8 is added to it, the sum is divided by two,
and twice the original number is subtracted from it, then the number is equal to the original
number.
Solution:
Let x be the original number.
Then,
A number is multiplied by 6: 6x
The product is added by 8: 6x + 8
The sum is divided by 2, it will be: 3x + 4
Twice of the original number will be subtracted from 3x + 4: x + 4
Subsequently, 4 is equal to 0 and I would be left with the original number, x.
5. Use deductive reasoning to complete the magic square below:
2 13 16 3
11 8 5 10
7 12 9 6
14 1 4 15
Using deductive reasoning, I had arrived in a conclusion that all row, column and
diagonal must add up to the same given number given the basic idea of a magic square. There
is no ideal row, column, or diagonal in the magic square above that is completely filled with
numbers. So, given that 34 is the smallest sum possible using the numbers 1 to 16 of a 4x4
magic square, I have no alternative but to believe that each row, column, and diagonal must
result in 34. As a result, after gathering all the missing numbers and adding them together, they
all equal to 34.
Activity 4.3
Directions: Solve the following problem comprehensively using the four steps in problem
solving devised by Polya. Show all your solutions, if necessary.
1. Four married are members of couples for Christ. The wives’ names are Kathy, Sally, Jovie,
and Apple. Their husbands are (in some order) Dannie, Wally, George, and Fernando.
• Wally is Jovie’s Brother.
• Jovie and Fernando dated some, but then Fernando met his present wife.
• Kathy is married to George
• Apple has two brothers.
• Apple’s husband in an only child. Find out who is married to whom.
My Solution:
o Wally is Jovie’s brother, so they cannot be married.
o Jovie and Fernando dated before, but then Fernando met his present wife and he
can’t be married to Kathy because Kathy is Married to George.
o Wally can’t be married to Kathy because she is married to George.
o Apple’s husband is an only child so Wally can’t be her husband because Wally
has a sibling.
o Fernando is the only man that can be married to Apple which is not related to
Wally nor Jovie.
o The only people left were Jovie, Dannie, Wally, and Sally.
o Since Wally is Jovie’s brother, Wally’s wife is Sally and Dannie’s wife is Jovie.
The Married Couples:
o George and Kathy
o Apple and Fernando
o Jovie and Dannie
o Wally and Sally
