1. How many thousands are there in the number 530070?

2. How many tens are there in the number 7805?

3. How many of the numbers are prime: 2; 23; 29; 46; 101; 119; 122.

4. Which of the expression is always odd for all values of 𝑛𝑛?

5. How many of the following pairs are co-prime numbers? 3, 4, 8, 13, 26.

6. How many co-prime numbers with 30 among the nubers not greater than 30?

7. How many co- prime numbers with 24 among the nubers not greater than 24?

8. The value of the expression (7𝑎𝑎 − 3)2 is odd (𝑎𝑎 ∈ 𝑁𝑁). Which of the followings is even number?
9. If 𝑥𝑥 and 𝑦𝑦 are even numbers, which of the followings can be an odd number?

10. If ����� ��� = 179, find 𝑎𝑎 + 𝑏𝑏.

11. If ���
𝑎𝑎𝑎𝑎 = 2𝑎𝑎 + 4𝑏𝑏, find 𝑎𝑎 + 𝑏𝑏.

12. If ������� 𝑥𝑥𝑥𝑥 = 𝑎𝑎 (𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑏𝑏), find 𝑎𝑎𝑎𝑎.

13. If 7𝑛𝑛 + 7 and 8𝑛𝑛 + 2 are consecutive even numbers, find the product of all possible values of 𝑛𝑛.

14. If 4𝑛𝑛 + 10 and 5𝑛𝑛 + 7 are consecutive odd numbers, find the sum of all possible values of 𝑛𝑛.

����−𝑦𝑦
𝑥𝑥𝑥𝑥
15. For the numbers 𝑥𝑥𝑥𝑥
��� and 𝑦𝑦𝑦𝑦
���, ����−𝑥𝑥 = 4, find the greatest value of 𝑥𝑥 + 𝑦𝑦.
𝑦𝑦𝑦𝑦
16. Match up:
��� + ��� a. is divisible by 9
1. 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏
b. is divisible by 37
��� − ���
2. 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏
c. is divisible by 11
3. ����� �����
𝑎𝑎𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏𝑏𝑏
d. is divisible by 100

e. is divisible by 111.

17. Work out: 54 ∙ 37 + 46 ∙ 37 − 54 ∙ 24 − 46 ∙ 24.

18. If the dividend is decreased 4 times, how to change the divisor in order to increase quotient 3 times?

19. If the dividend is increased 5 times, how to change the divisor in order to decrease quotient 3 times?

20. Work out: 11011:1001

21. Work out: 22022:2002.

22. If the dividend is increased 18 times, how to change the divisor in order to decrease quotient 3 times?
23. If the sum of the ages of Togrul and Arif will be 29 after 5 years, what is the sum of their ages at the
present?

24. Difference is 147 less than minuend. Find the subtrahend.

25. Find the difference of the greatest and the smallest five digit number whose digits are different.

26. If 1 + 3 + 5 + ⋯ + 19 = 𝑎𝑎, what is the value of 2221 + 2223 + 2225 + ⋯ + 2239 in terms of 𝑎𝑎?

27. If 3 + 7 + 11+. . . +27 = 𝑎𝑎, what is the value of 233 + 237 + 241 + ⋯ + 257 in terms of 𝑎𝑎 ?

28. For 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 − natural numbers 𝑎𝑎𝑎𝑎 = 19, 𝑏𝑏𝑏𝑏 = 7. Find 𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐.

29. For 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 − natural numbers 7𝑦𝑦 = 8𝑧𝑧, 𝑥𝑥 = 2𝑧𝑧. Find the minimum value of 𝑥𝑥 + 𝑦𝑦 + 𝑧𝑧.
30. For 4𝑥𝑥 − 3 and 3𝑦𝑦 + 2 are co-prime numbers, 22(4𝑥𝑥 − 3) = 58(3𝑦𝑦 + 2), find 𝑥𝑥𝑥𝑥 (𝑥𝑥, 𝑦𝑦 ∈ 𝑁𝑁)

31. For 5𝑥𝑥 − 4 and 3𝑦𝑦 + 2 co-prime numbers, 58(5𝑥𝑥 − 4) = 22(3𝑦𝑦 + 2), find 𝑥𝑥𝑥𝑥 (𝑥𝑥, 𝑦𝑦 ∈ 𝑁𝑁)

32. When the number is divided by 17, partial quotient is 3 and reminder is 16. Find this number.

33. When the number is divided by 19, partial quotient is 3 and reminder is 14. Find this number.

34. Find the reminder when the number 𝑛𝑛 = 1 ∙ 2 ∙ 3 ∙ 4 ∙ 5 ∙ 6 ∙ 7 ∙ 8 ∙ 9 ∙ 10 + 1 is divided by 15.

35. Find the reminder when the number 𝑛𝑛 = 1 ∙ 2 ∙ 3 ∙ 4 ∙ 5 ∙ 6 ∙ 7 ∙ 8 ∙ 9 ∙ 10 + 1 is divided by 18.

�������� is divisible by 3. Which of the following cannot be the value for 𝑎𝑎 + 𝑏𝑏?
36. The five digit number 4𝑎𝑎5𝑏𝑏7

37. When the number 𝑎𝑎 is divided by 10, quotient is 𝑐𝑐 and reminder is 8. What is the quotient when this
number is divided by 5?

38. When the number 𝑎𝑎 is divided by 8, the reminder is 7. What is the quotient when the number 2𝑎𝑎 + 1 is
divided by 8?

39. When the natural number is divided by 21, the reminder is 16. What is the reminder when this number is
divided by 7.

40. When the natural number is divided by 16, the reminder is 15. What is the reminder when this number is
divided by 4.
 ���������� is divisible by 6, what is the sum of all possible values of 𝑎𝑎?
41. If the number 52438𝑎𝑎

42. What is the reminder when the product 147776 ∙ 147773 is divided by 7?

43. What is the reminder when the product 328885 ∙ 328882 is divided by 8?

44. What is the reminder when the number 𝑎𝑎 = (7219 ∙ 35274)2 + 3 is divided by 4?

45. What is the reminder when the number 𝑎𝑎 = (7506 ∙ 16551)2 + 4 is divided by 25?

��� is divided by 9 the reminder is 5. Find 𝑐𝑐 if the number 𝑎𝑎𝑎𝑎𝑎𝑎

47. What is the number of different prime factors of 420?

48. For which values of 𝑛𝑛, 𝐻𝐻𝐻𝐻𝐻𝐻(𝑛𝑛; 10) = 𝑛𝑛?

49. The length of Vagif’s and Sabir’s steps are accordingly 80 cm and 70 cm. In which shortest distance the
number of their steps will be equal?

50. Which of the followings could not be reminder when divisor is 8?

51. Find the number of different prime factors of 𝑎𝑎 = 6 ∙ 8 ∙ 35?

A) 𝑏𝑏 B) 1 C) 𝑎𝑎 + 𝑏𝑏 D) 𝑎𝑎𝑎𝑎 E) 𝑎𝑎

53. If 𝑎𝑎 is the divisor of 𝑏𝑏, find 𝐻𝐻𝐻𝐻𝐻𝐻(𝑎𝑎; 𝑏𝑏).

54. If 𝑎𝑎 is the divisor of 𝑏𝑏, find 𝐿𝐿𝐿𝐿𝐿𝐿(𝑎𝑎; 𝑏𝑏).

55. If 𝑎𝑎 and 𝑏𝑏 are co-prime numbers, find 𝐿𝐿𝐿𝐿𝐿𝐿(𝑎𝑎; 𝑏𝑏).

𝐿𝐿𝐿𝐿𝐿𝐿(𝑎𝑎;𝑏𝑏)
56. If 𝑎𝑎 and 𝑏𝑏 are co-prime numbers greater than 1, find .
𝐻𝐻𝐻𝐻𝐻𝐻(𝑎𝑎;𝑏𝑏)

57. If 𝑎𝑎 is the divisor of 𝑏𝑏, find 𝐿𝐿𝐿𝐿𝐿𝐿(𝑎𝑎; 𝑏𝑏) − 𝐻𝐻𝐻𝐻𝐻𝐻(𝑎𝑎; 𝑏𝑏).

58. When the number is divided by 12, 15, and 20 the reminder is 3. Find the smallest number that satisfies
this condition.

59. When the number is divided by 5, 6, and 7 the reminder is 2. Find the smallest number that satisfies this
condition.
60. When students lined up there were 9 students in one row, then they lined up again and this time there
were 15 people in one row. If the number of all students is more than 120 and less than 150, How many
students are there?

61. There 62 boys and 93 girls in a competition. The combination of boys and girls in one team is same. How
many girls in one team?

𝑎𝑎𝑎𝑎
62. 𝑎𝑎 and 𝑏𝑏 are natural numbers, 𝑐𝑐 is the product of prime factors which are not common. If = 256, find
𝑐𝑐
𝐻𝐻𝐻𝐻𝐻𝐻(𝑎𝑎; 𝑏𝑏).

𝑛𝑛 𝑛𝑛 𝑛𝑛
63. If the values of the expressions , , are natural numbers, what is the sum of the digits of the smallest
6 8 9
value of 𝑛𝑛?

𝑛𝑛 𝑛𝑛 𝑛𝑛
64. If the values of the expressions , , are natural numbers, what is the sum of the digits of the
12 18 10
smallest value of 𝑛𝑛?

65. From a point two objects started moving along the circles at the same time. First object reaches the
starting point every 50 seconds and the second object reaches every 60 seconds. In which shortest time will
both of the point be at the starting point?

66. The product of the numbers 𝑎𝑎 and 𝑏𝑏 is 72000, if the product of prime factors which are not common is
20. Find 𝐿𝐿𝐿𝐿𝐿𝐿(𝑎𝑎; 𝑏𝑏).
67. The product of the numbers 𝑎𝑎 and 𝑏𝑏 is 32000, if the product of prime factors which are not common is
20. Find 𝐿𝐿𝐿𝐿𝐿𝐿(𝑎𝑎; 𝑏𝑏).

68. If 𝑎𝑎 = 4 ∙ 3𝑚𝑚+2 , 𝑏𝑏 = 12𝑚𝑚 (𝑚𝑚 ∈ 𝑁𝑁) and 𝐻𝐻𝐻𝐻𝐻𝐻(𝑎𝑎; 𝑏𝑏) = 108, find the value of 𝑎𝑎.

69. The sum of 𝐿𝐿𝐿𝐿𝐿𝐿 and 𝐻𝐻𝐻𝐻𝐻𝐻 of two even consecutive numbers is 222. Find the smallest number.

70. The sum of 𝐿𝐿𝐿𝐿𝐿𝐿 and 𝐻𝐻𝐻𝐻𝐻𝐻 of two even consecutive numbers is 422. Find the smallest number.

𝑎𝑎 3
71. For natural numbers 𝑎𝑎 and 𝑏𝑏, = and 𝐿𝐿𝐿𝐿𝐿𝐿(𝑎𝑎; 𝑏𝑏) + 𝐻𝐻𝐻𝐻𝐻𝐻(𝑎𝑎; 𝑏𝑏) = 104. Find 𝑏𝑏 − 𝑎𝑎.
𝑏𝑏 4

72. A master can prepare 3 chairs in a day and, his assistant 2 chairs. In a few days they prepared 60 chairs
together. How many of these chairs were prepared by the master’s assistant?

73. Along the border of a small rectangular park with the length of 64 meter and the width of 48 meter light
poles at the same distances from each other were installed. The cost of each light pole is 40 manat. At least
how much money is spent on purchasing light poles?
74. In the diagram the different prime factors and the number of these prime factors of the composite
numbers m and n are given. Write the prime factorization of m and n and find the ratio

LCM(m,n): HCH(m,n).

The number of prime factors

Different prime factors

75. The prime factorization of the numbers a and b are given below.

𝑎𝑎 = 2𝑚𝑚 ∙ 35 ∙ 7𝑛𝑛

𝑏𝑏 = 23 ∙ 3𝑘𝑘 ∙ 74 ∙ 52
If 𝐻𝐻𝐻𝐻𝐻𝐻(𝑎𝑎, 𝑏𝑏) = 22 ∙ 34 ∙ 73 , then find the product 𝑚𝑚 ∙ 𝑛𝑛 ∙ 𝑘𝑘

76. Along the border of a triangular garden with the lengths of sides 48m, 60m and 60m trees at the same
distance away from each other should be planted. At least how many trees are needed?