# Solutions

# Problems and Pythagoras numbers

### Calculate the divisions below with your mind and mark the fair ones with an x :

1. 48/2 x

2. 48/3 x

3. 48/4 x

4. 48/5

5. 48/6 x

6. 48/7

7. 48/8 x

8. 48/9

1. 45/2

2. 45/3 x

3. 45/4

4. 45/5 x

5. 45/6

6. 45/7

7. 45/8

8. 45/9 x

### Now write below all the divisors of:

· 48: ……● 1, 2, 3, 4, 6, 8, 12, 16, 24, 48…………

· 45: ……1, 3, 5, 9, 15, 45…………………………

### Problem:

I have a collection of 225 stamps and I want to place them in an album. Every page of it has space for 30 stamps maximum. How many stamps can I place in each page so I can use the fewer of them and have the same number of stamps in each one?

Solution: 225 divisors: 1, 3, 5, 9, 25, 45, 75, 225

I have to place 25 stamps in each page as it’s the biggest number from 225 divisors that is also <30.

### Circle the numbers that are divided with 2, 4 and 9 at the same time:

100 302 815 150 925 300

**3600 8136 ** 8082 1306 5127 9246

### Write down each one of the numbers below as a product of two factors:

● 10: 2x5

● 35: 7x5

● 48: 6x8

● 54: 6x9

● 63: 3x21

● 72: 6x12

● 81: 9x9

● 93: 3x31

### Calculate with your mind and write down each one of the numbers below as a product of prime factors:

● 10: 2x5

● 30: 3x2x5

● 50: 2x5x5

● 70: 2x5x7

● 20: 2x2x5

● 40: 2x2x2x5

● 60: 2x2x3x5

● 80: 2x2x2x2x5

### Circle the correct one:

(LCM: Least / Lowest Common Multiple)

· LCM(4, 9): a. 9 b.18 c. 27 **d.36** e.72

· LCM(10, 15): a. 15 b. 20 **c. 30** d. 60 e. 150

· LCM(7, 35): ** a. 35** b. 70 c. 105 d. 245 e. 700

### Problem:

Three friends went to the park with their bikes. They started together the cycling of the trail. It took the first one 4 minutes to finish one round, the second one 6 minutes and the third one 8 minutes. In how many minutes will they pass together from the same spot and how many rounds will each one have made?

Solution: LCM (4, 6, 8): 24

They’ll pass together from the same spot in 24 min.

The first one: 24/4= 6 rounds

The second one: 24/6= 4 rounds

The third one: 24/8= 3 rounds

### Problem:

Katherine practices the trumpet every 11th day and the flute every 3rd day.

Katherine practiced both the trumpet and the flute today.

How many days is it until Katherine practices the trumpet and flute again on the same day?

Solution: LCM (3, 11): 33 days

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, __33__

Multiples of 11: 11, 22, __33__

### Problem:

Paul goes golfing every 6th day and Nikos goes golfing every 7th day.

If Paul and Nikos both went golfing today, how many days is it until they go golfing on the same day again?

Solution: LCM (6,7): 42 days

### Problem:

Anastasia and George ended up making the same number of biscuits for a bake sale at school, even though Anastasia made them in batches of 7 biscuits and George made them in batches of 11 biscuits. What is the smallest number of biscuits each one must have baked?

Solution: LCM (7, 11): 77 biscuits

### Problem:

Marina baked 30 oatmeal cookies and 48 chocolate chip cookies to package in plastic containers for her friends at school. She wants to divide the cookies into identical containers so that each container has the same number of each kind of cookie. If she wants each container to have the greatest number of cookies possible, how many plastic containers does she need?

Solution: The greatest common divisor of 30 and 48 is 6, so she needs six containers. Each one will include 30/6= 5 oatmeal cookies and 48/6= 8 chocolate chip cookies.

### Suppose we have a rectangular triangle ABC where AB=5, BC=12 and AC is the hypotenuse. Find the AC side.

Solution: From the Pythagorean Theorem:

AC^{2}= AB^{2 }+ BC^{2}= 5^{2 }+ 12^{2}= 25 + 144= 169

AC= 13

### Suppose we have a triangle ABC where ΑΒ=5 cm, BC=3 cm and B=90°. How many cm is the AC side?

Solution: AC^{2}= AB^{2 }+ BC^{2}= 5^{2} + 3^{2}= 25 + 9= Square root of 34 cm

### Prove that each point of the perpendicular bisector of a line segment is equidistant from the ends.

Solution:

We compare the two triangles AMP, BMP:

AM=MB

M=90⁰

PM is a common side

So the two triangles are equal and the perpendicular bisector of the line segment is equidistant from the ends.

### In the figure below prove that AB is Line bisector of KL when the two circles are equal.

Solution: AK=AL=AB=AL=R

So, as long as the upper equation is true, AB is the line segment of KL.

### If the red lines represent the distances between Denmark, Greece, Spain and France, where do we have to meet in the summer to be in an equidistant place?

Solution: If we bring the line bisector of every line, we’ll find the common point. That point will be our meeting place.