Number and Place Value Math Olympiad

Number System in details for Math Olympiad 

Numbers

Numerical value that is use to represent anu data, quality, measurement and way of counting

 (like 1,2,3,4......... ). 

Natural  Number

Natural number the number that start with 1 till infinity. It represents by 'N' (Not n).

Example: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17......

Odd Natural Number 

All Natural numbers which are not divided by 2 are called Odd numbers 

Example 

1,3,5,7,9,11...............

Even   Natural Numbers 

All Natural numbers which are divided by 2 

Example 

2,4,6,8,10.......

Whole Numbers 

All Natural counting numbers along with zero (0) are called Whole Numbers, it represents by 'W' ( Not w) 

Example 

0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15........

Integers 

The set of all counting Natural Number along with 0 and all negative Number without any fraction and Decimals numbers are called Integers. It represents by 'Z' (Not z)

Example 

(........-5,-4,-3,-2,-1,0,1,2,3,4,5,6......)

Negative Integers 

All Integers value less than zero(0) are called Negative Integers, they always come with Negative sign.

(....-10,-9,-8,-7.....-1)

Positive Integers 

All Integers greater than zero(0) are called Positive they may write with + sign or without sign.

Example

+1,+2,+3,+4,+5,+6,+7,+8..........

Or

1,2,3,4,5,6,7,8,9.........

Zero Integer

It contains only   zero 

Example 

0

What is Place Value?

The expanded form of the digit

4567802 = 

4×1000000+5×100000+6×10000+7×1000+8×100+0×10+2×1

Write these numbers in expanded form in same way 

2675087

2234561

459876

3431001

What is Face Value ?

A face value of a number always same it never changes like place value 

For example, in digit 2573078

Place values of 7 are two 70000 and 70 but face value of 7 always 7. 

How to Write a digit in International System?

Here in this chart 

8 is in one Place i.e.8

3 is in 10 Place I.e. 30

6 is in 100 Place I.e.600

1 is in 1000 Place i.e.1000

7 is in 10000 Place i.e 70000

4 is 100000 Place i.e 400000

2 is in 1000000 Place i.e 2000000

9 is in 10000000 Place i.e 90000000

Seven Digit Numbers

1000897 One Million Eight Hundred Ninety-Seven

2078901 Two Million Seventy-Eight Thousand Nine Hundred One.

3002312 Three Million Two Thousand Three Hundred Twelve 

4000125 Four Million One Hundred Twenty Five 

8650000 Eight Million Six Hundred Fifty Thousand 

4524561 Four Million Five Hundred Twenty-Four Thousand 5 hundred Sixty-One.

Eight Digit Numbers

70,000,000 - Seventy million

40,000,030 - Forty million thirty

60,000,047 - Sixty million forty-seven

 20,000,123 - Twenty million one hundred twenty-three

 80,009,000 - Eighty million nine thousand

50,049,050 - Fifty million forty-nine thousand fifty

 90,200,000 - Ninty million two hundred thousand

 30,301,000 - Thirtty million three hundred one thousand

 60,500,070 - Sixty million five hundred thousand seventy

Successor and Predecessor

Successor and Predecessor are originated from 'Succeed' and 'Precede'.

A Successor is a number that comes just after Given number it means whatever the number given add 1 to get it Successor 

For Example, 

Successor of 27 will be 28 (27+1)

Successor of 49 will be 50 (49+1)

Successor of 99 will be 100 (99+1)

A Predecessor is a number that comes just before given number we minus 1 from the given number to get Predecessor.

For Example 

Predecessor of 27 is 26(27-1)

Predecessor of 49 is 48 (49-1)

Predecessor of 99 is 98 (99-1)

Roman Numbers 

The Roman numeral system is an ancient method of representing numbers that remains in use today. Instead of using standard Arabic numerals like 1, 2, 3, 4, and 5, it employs letters from the Latin alphabet (like I, V,X,L,M etc.)to create unique numerical representations.

Factors 

Factors are numbers obtained by dividing another number completely without leaving any remainder

Example 

6÷3=2 means 3 is a factor of 6. 

6÷4 Here we get reminder 2 after dividing 6 by 4 so 4 is not a factor of 6. 

Multiples

When we multiply a particular number to another number then we get multiples 

For example 

8×1=8

8×2=16

8×3=24

8×4=32

8×5=40 

Continue.

The multiples of 8 are 8,16,24,32,40........

Prime Factorization :

Prime factorization is the process of expressing a number as a product of prime numbers.  

A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself.  They are not divided by any other number except 1 and itself. Between 1 and 100,there are 25 prime numbers.

2 is only even prime number.

Other examples of Prime Numbers 

3 ,5,7,11,13,17...

Don't confuse all odd numbers are not prime numbers.

Common Multiples and Common Factors

Common Multiples

Multiples that are shared by two or more numbers are called their common multiples. For example:

Multiples of 6: 6,12,18,24,30,36,42,48,54,60,66,72....

Multiples of 8 :

8,16,24,32,40,48,56,64,72,80,88,96....

Common Multiples are 

24,48,72,

Common Factors :

Common factors of any two numbers are the factors which appear in both the numbers 

Example 

Factors of 20 are 1,2,4,5,10,20

Factors of 30 are 1,2,3,5,6,10,15,30

Common Factors of 20 and 30 are 1, 2 ,5 and 10 

10 is the highest factor in 1,2,5 and 10 so Highest Common Factor is 10.

What is HCF, GCF, or GCD?

The Highest Common Factor (HCF) also known as the Greatest Common Factor (GCF) or Greatest Common Divisor (GCD), is the largest positive number that divides all the given numbers evenly.  

In simple terms, it is the greatest number that is a factor of each of the given numbers.  

For example, consider the numbers 24 and 30

- Factors of 24  1, 2, 3, 4,6,12 and 24

- Factors of 30- 1,2,3,5,6,10,15,and 30

- Common factors: 1,2,3,6 

- The HCF (GCF or GCD) is 6 as it is the largest common factor.  

Understanding HCF is important for simplifying fractions, solving number problems, and various mathematical applications.  

Division   Method to Find The HCF 

How to Find the Highest Common Factor (HCF) Using the Division Method

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest number that divides two or more numbers without leaving a remainder. One of the most efficient ways to determine the HCF of two numbers is by using the division method. Follow these step-by-step instructions to find the HCF.

Steps to Find HCF Using the Division Method

Divide the larger number by the smaller number – Begin by dividing the greater number by the smaller one.

Check the remainder – If the remainder is zero, the divisor is the HCF. Otherwise, continue to the next step.

Repeat the process – Take the remainder from the previous division and divide the previous divisor by this remainder.

Continue until you get zero as the remainder – The last non-zero remainder before reaching zero is the HCF of the given numbers.

Example

Find the HCF of 48 and 18 using the division method:

Step 1: Divide 48 by 18 → 48 ÷ 18 = 2, remainder 12

Step 2: Divide 18 by 12 → 18 ÷ 12 = 1, remainder 6

Step 3: Divide 12 by 6 → 12 ÷ 6 = 2, remainder 0

Since the remainder is now zero, the last divisor (6) is the HCF of 48 and 18.

Why Use the Division Method?

It is a fast and efficient way to find the HCF.

Works well for large numbers compared to prime factorization.

Helps in simplifying fractions and solving LCM-HCF problems.

By mastering the division method for HCF, students can solve complex mathematical problems more easily. Try practicing with different numbers to strengthen your problem-solving skills!

Prime Factorization Method to Find HCF 

I. Find the All factors of given numbers 

II. Find out all Common factors.

III. Calculate the product of these Common factors which must be equal to Highest Common Factor. 

HCF by Division Method 

Factor (H.C.F.) using the division method 

1. Divide the larger number by the smaller number.  

2. If there is a remainder, divide the previous divisor (the smaller number) by this remainder.  

3. Repeat the process, using the remainder as the new divisor and the previous divisor as the new dividend.  

4. Continue until the remainder becomes 0.  

5. The last divisor before reaching 0 is the H.C.F. of the given numbers.

Lowest Or Least Common Multiples (LCM)

LCM of two or more than 2 numbers is smallest Positive numbers that is divided by given numbers.

For Example 

Find the LCM of 12 and 30

Factors of 12 

12=1×2×2×3

30×1×2×3×5

LCM : 2×2×3×5 = 60.

So 60 is the smallest Positive number that is divided by 12 and 30 both.

We can find LCM by two method  

i. Division Method 

Write all numbers together by using comma , divide by numbers till we get 1 at last.

Multiply all all factors

Rounding Up of Numbers 

Rounding up is a method used to estimate a number by adjusting it to the nearest value that makes sense in a given situation.  

How to round numbers to the nearest tens  

look at the digit in the one's place, which is the rightmost digit. if this digit is between 0 and 4, round down by keeping the tens place digit the same and changing all digits to the right to 0. if the one's place digit is between 5 and 9, round up by adding 1 to the tens place digit and replacing all digits to the right with 0.

Practice Questions for IMO Math Olympiad 

100 number system questions for Math Olympiad practice, organized by topic. These questions cover concepts from Class 3 to Class 8 and range from basic to advanced levels.  

i. Place Value and Face Value (10 Questions)

1. Write the place value of 5 in 25,678.  

2. Find the face value of 9 in 9,876.  

3. What is the sum of the place values of all digits in 4,325?  

4. Write the expanded form of 74,309.  

5. Identify the digit at the thousands place in 98,765.  

6. How many times is the place value of 7 in 7,451 greater than its face value?  

7. Write the smallest 5-digit number using 1, 3, 5, 7, and 9 without repetition.  

8. Find the difference between the place value and face value of 6 in 6,403.  

9. If the place value of a digit is 80,000 and the digit is 8, what is the number?  

10. Replace * in 1*56 with a digit so that the number is divisible by 5.  

ii.  Divisibility Rules (10 Questions)

11. Is 4,752 divisible by 3?  

12. Check if 7,842 is divisible by 9.  

13. Find the smallest number that is divisible by both 6 and 8.  

14. A number is divisible by 11 if the difference of the sum of its alternating digits is a multiple of 11. Check if 62,347 is divisible by 11.  

15. Find the missing digit * in 45*32 so that it is divisible by 4.  

16. A number ends with 0. Name three divisibility rules that confirm it is divisible by those numbers.  

17. Check if 987,654 is divisible by 6.  

18. What is the smallest 4-digit number divisible by 7?  

19. Find a 3-digit number divisible by both 5 and 9.  

20. A number is divisible by both 8 and 12. What is the smallest such number?  

iii. Prime and Composite Numbers (10 Questions)

21. Is 1 a prime or composite number? 

 

22. List all prime numbers between 50 and 80.  

23. Find the sum of the first five prime numbers.  

24. How many prime numbers are there between 1 and 100?  

25. A number is divisible by only 1 and itself. What type of number is it?  

26. Identify the smallest composite number.  

27. Find the largest prime number less than 100.  

28. Express 77 as a product of prime factors.  

29. Is 119 a prime number? Justify your answer.  

30. Find the prime factorization of 144.  

iv. LCM and HCF (10 Questions)

31. Find the LCM of 18 and 24.  

32. Calculate the HCF of 45 and 60.  

33. Find two numbers whose LCM is 72 and HCF is 12.  

34. If the HCF of two numbers is 6 and their product is 216, find their LCM.  

35. The LCM of two numbers is 120 and their HCF is 10. Find the product of the numbers.  

36. Find the HCF of 36, 48, and 60.  

37. If the LCM of two prime numbers is 143, find the numbers.  

38. The sum of two numbers is 72, and their HCF is 6. Find their LCM.  

39. Find the smallest number divisible by both 15 and 20.  

40. Two numbers are co-prime. What is their HCF?  

v. Properties of Numbers (10 Questions)

41. Find the sum of the first 20 natural numbers.  

42. The sum of any two odd numbers is always ____.  

43. The product of an even and an odd number is always ____.  

44. Find the sum of the first 50 even numbers.  

45. If a number is divisible by 4 and 9, must it be divisible by 36? Explain.  

46. If a number is multiplied by 1, what will be the result?  

47. What is the sum of the smallest odd and smallest even prime number?  

48. Find the product of all prime numbers less than 10.  

49. Is the sum of two prime numbers always even? Give an example.  

50. How many even numbers are there between 1 and 100?  

vi. Fractions and Decimals (10 Questions)

51. Convert 0.75 into a fraction.  

52. Express 7/8 as a decimal.  

53. Find the sum of 3/5 and 4/7. 

54. Subtract 2.35 from 5.8.  

55. Multiply 2/3 by 5/6 and simplify.  

56. Divide 4.5 by 0.9.  

57. Arrange in ascending order: 3/4, 2/3, 5/6.  

58. Convert 3.125 into a fraction.  

59. Find 25% of 480.  

60. What is 0.2 × 0.4?  

vii. Squares, Cubes, and Roots (10 Questions)

61. Find the square of 18.  

62. What is the cube of 7?  

63. Find the square root of 144.  

64. Find the cube root of 27.  

65. What is the sum of squares of 4 and 5?  

66. If x² = 64, find x.  

67. Find the smallest perfect square greater than 90.  

68. Find the largest perfect cube less than 500.  

69. The square root of which number is 15?  

70. Find two consecutive numbers whose squares differ by 25.  

viii. Rational and Irrational Numbers (10 Questions)

71. Is √16 a rational or irrational number?  

72. Identify whether 22/7 is rational or irrational.  

73. Express 0.333... as a fraction.  

74. Find three rational numbers between 1 and 2.  

75. Find the decimal expansion of 7/11.  

76. Is 3.141592 a rational number?  

77. Identify the irrational number: √2, 4/5, 0.25, 1.75.  

78. Find a rational number between 2/3 and 3/4.  

79. What is the sum of two rational numbers?  

80. What happens when a rational and an irrational number are added?  

ix. Number Puzzles and Logical Reasoning (10 Questions)

81. A number is 3 more than twice another number. Their sum is 27. Find the numbers.  

82. A three-digit number has 5 in the hundreds place, 4 in the tens place, and the sum of all digits is 12. Find the number.  

83. Find the missing number: 2, 6, 12, 20, __, 42.  

84. A number leaves a remainder of 2 when divided by 5. What could be the number?  

85. Find the number which is both a perfect square and a perfect cube.  

86. Solve: 5x - 7 = 3x + 9.  

87. The sum of two numbers is 50, and their difference is 10. Find the numbers.  

88. Find the missing digit: 3_6 × 3 = 1098.  

89. A number is divisible by both 2 and 3. Name another number it must be divisible by.  

90. The product of two numbers is 72, and their LCM is 36. Find their HCF.  

x. Mixed Problems 

91.What is the sum of the first 15 odd numbers?

92.If a number is divisible by both 9 and 12, what other number must it be divisible by?

93.Find the greatest 4-digit number that is divisible by 7.

94.A number is 6 times its unit digit. If the sum of its digits is 9, what is the number?

95.Find the smallest number that has exactly 8 factors.

96.How many numbers between 1 and 100 are divisible by both 2 and 5?

97.Find a two-digit prime number whose sum of digits is 10.

98.If 5x + 3 = 2x + 12, find x.

99.Find the least number that must be added to 3456 to make it a multiple of 11.

100.What is the sum of all factors of 36?