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Cambridge IGCSE0607

Types of number

International Mathematics 0607 Chapter Notes

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Types of number
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In this chapter

  1. 1Natural numbers, integers and words
  2. 2Square, cube, triangle and prime numbers
  3. 3Factors, multiples and prime factors
  4. 4Highest common factor (HCF)
  5. 5Lowest common multiple (LCM)
  6. 6Reciprocals

1. Natural numbers, integers and words

Natural numbers are 0, 1, 2, …. Integers are positive, zero and negative whole numbers.

A rational number can be written as a fraction. An irrational number cannot: π is one. A number written in fraction form, or a decimal that stops, is not irrational.

To convert between words and figures, read the place of each digit. Six billion is 6 000 000 000. Ten thousand and seven is 10 007.

Examiner insight

“A surprising number of candidates omitted the number –3 from their list of integers, perhaps forgetting that integers can be negative as well as positive.” “The fraction or the decimal number were frequently included in the list of irrational numbers.”

Worked example 12 marks

(b) Write the number fifty-five thousand two hundred in figures.

(c) Write down the place value of 3 in the number 234 100.

  1. 1

    fifty-five thousand = 55 000, two hundred = 200, so 55 200 [1]

  2. 2

    the 3 is in the ten thousands column, so its place value is 10 000 (or 30 000) [1]

2. Square, cube, triangle and prime numbers

A square number is a whole number multiplied by itself: 1, 4, 9, 16, 25, 36, 49, 64. A cube number is a whole number used three times in a product, like 3 × 3 × 3 = 27: 1, 8, 27, 64.

A triangle number comes from adding 1, then 2, then 3 and so on: 1, 3, 6, 10, 15, 21, 28.

A prime number has exactly two factors, 1 and itself. So 1 is not prime, and 2 is the only even prime.

Examiner insight

“Only a small minority of candidates knew what a triangle number was.” “Part (v) was found the most difficult, where 1 was often seen as a prime number.”

Worked example 13 marks

x and y are integers between 26 and 65.

x is a square number, y is a cube number, (x -y) is a prime number.

Find the value of x and the value of y.

  1. 1

    squares between 26 and 65: 36, 49, 64. Cubes: 27, 64 [1]

  2. 2

    try the differences: 36 - 27 = 9 and 49 - 27 = 22 are not prime; 64 - 27 = 37 is prime [1]

  3. 3

    x = 64 and y = 27 [1]

3. Factors, multiples and prime factors

A factor of a number divides into the number without leaving a remainder. A multiple of a number is in that number's times table. Factors of a number are smaller than or equal to it. Multiples are larger than or equal to it.

To write a number as a product of its prime factors, divide by primes until only primes are left. A factor tree does the same job.

Write repeated primes with powers: 72 = 2³ × 3².

Examiner insight

“The majority of candidates drew a factor tree and then wrote the product of prime factors correctly.”

Worked example 12 marks

Express 72 as a product of its prime factors.

  1. 1

    72 = 2 × 36 = 2 × 2 × 18 = 2 × 2 × 2 × 9 [1]

  2. 2

    72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3² [1]

4. Highest common factor (HCF)

The highest common factor of two numbers is the largest number that divides into both. It is one value.

Write each number as a product of primes. Multiply the primes that appear in both lists, using the lower power of each.

The HCF is a factor, so it is not larger than either number.

Examiner insight

“As 5 and 7 are both prime numbers, their HCF is 1. Some multiplied the numbers to get 35, the lowest common multiple or even 70, the next common multiple.” “Students should remember that for the HCF, the answer is a factor, so smaller than the given numbers.”

Worked example 12 marks

Find the highest common factor (HCF) of 180 and 96.

  1. 1

    180 = 2² × 3² × 5 and 96 = 2⁵ × 3 [1]

  2. 2

    common primes: 2² × 3 = 12 [1]

5. Lowest common multiple (LCM)

The lowest common multiple of two numbers is the smallest number that is a multiple of both.

List the multiples of each until one appears in both lists. Or take every prime that appears in either number, using the higher power of each.

The LCM is a multiple, so it is not smaller than either number.

Examiner insight

“Candidates should remember that for the LCM, the answer will be a multiple, so larger than the given numbers.”

Worked example 12 marks

Find the lowest common multiple (LCM) of 35 and 42.

  1. 1

    35 = 5 × 7 and 42 = 2 × 3 × 7 [1]

  2. 2

    LCM = 2 × 3 × 5 × 7 = 210, using the 7 once [1]

6. Reciprocals

The reciprocal of a number is 1 divided by that number. For a fraction, turn it upside down.

Write a decimal as a fraction first when that is quicker. A fraction and its decimal form are both accepted.

reciprocal of a = 1/a

reciprocal of a/b = b/a

Examiner insight

“Only a minority of candidates knew the word reciprocal.”

Worked example 12 marks

(a) Write 0.17 as a fraction.

(b) Write down the reciprocal of 5/16.

  1. 1

    0.17 = 17/100 [1]

  2. 2

    reciprocal = 16/5 (or 3.2) [1]

Recap

  • Natural numbers start at 0. Integers include negatives.
  • Square, cube and triangle numbers: 1, 4, 9 | 1, 8, 27 | 1, 3, 6, 10, 15.
  • A prime has exactly two factors. 1 is not prime.
  • Prime factors: divide by primes. Use powers for repeats.
  • HCF: common primes, lower powers. It is a factor, so it is small.
  • LCM: all primes, higher powers. It is a multiple, so it is large.
  • Reciprocal of a/b is b/a.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Write seven thousand and nine in figures.1 mark
  2. Here is a list of numbers: 5, -2, π, 0, 1/2, 8. (a) Write down all the integers in the list. (b) Write down the irrational number.3 marks
  3. Do not use a calculator. Here is a list of numbers: 9, 15, 21, 27, 37, 49. From the list, write down (a) the prime number (b) the cube number (c) the triangle number between 12 and 18.3 marks
  4. Do not use a calculator. Write down all the factors of 30.2 marks
  5. Do not use a calculator. Express 90 as a product of its prime factors.2 marks
  6. Do not use a calculator. Find the highest common factor of 36 and 60.2 marks
  7. Do not use a calculator. Find the lowest common multiple of 12 and 18.2 marks
  8. Write down the reciprocal of (a) 7/12 and (b) 0.2.2 marks
  9. A florist has 84 roses and 126 tulips. She uses all the flowers to make identical bunches. Every bunch has the same number of roses and the same number of tulips. (a) Find the greatest number of bunches she can make. (b) Write down the number of roses and the number of tulips in each bunch. (c) Roses are delivered every 8 days and tulips every 14 days. Both are delivered today. After how many days are both next delivered on the same day?5 marks
  10. Do not use a calculator. Two whole numbers have a highest common factor of 12 and a lowest common multiple of 360. One of the numbers is 72. Find the other number.4 marks

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