Number Theory and problems

1. 1
 Calculate the divisions below with your mind and
mark the fair ones with a :
1. 48/2
2. 48/3
3. 48/4
4. 48/5
5. 48/6
6. 48/7
7. 48/8
8. 48/9
1. 45/2
2. 45/3
3. 45/4
4. 45/5
5. 45/6
6. 45/7
7. 45/8
8. 45/9
Now write below all the divisors of:
 48: …………………………………………………………………
 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

2. 2
each page so I can use the fewer of them and have
the same number of stamps in each one?
Solution:
 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

3. 3
 Write down each one of the numbers below as a
product of two factors:
 10:
 35:
 48:
 54:
 63:
 72:
 81:
 93:
 Calculate with your mind and write down each one
of the numbers below as a product of prime
factors:
 10:
 30:
 50:
 70:
 20:
 40:
 60:
 80:

4. 4
 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:

5. 5
 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:
 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:

6. 6
 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:
 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:

7. 7
 Suppose we have a rectangular triangle ABC where
AB=5, BC=12 and AC is the hypotenuse. Find the AC
side.
Solution:
 Suppose we have a triangle ABC where ΑΒ=5 cm,
BC=3 cm and B=90°. How many cm is the AC side?
Solution:

8. 8
 Prove that each point of the perpendicular bisector
of a line segment is equidistant from the ends.
Solution:

9. 9
 In the figure below prove that AB is Line bisector of
KL when the two circles are equal.
Solution:

10. 10
 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?

11. 11
Solution: