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Cambridge IGCSE0580

Types of number

Mathematics 0580 Chapter Notes

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Types of number
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In this chapter

  1. 1Natural numbers, integers and number words
  2. 2Prime numbers
  3. 3Square numbers and cube numbers
  4. 4Factors, multiples and common factors
  5. 5Prime factors
  6. 6HCF and LCM
  7. 7LCM in a timetable problem
  8. 8Rational, irrational and reciprocals

1. Natural numbers, integers and number words

Natural numbers are the counting numbers, such as 3 and 250. A negative number is not a natural number. Integers are whole numbers that can be positive, zero or negative, such as -7, 0 and 12.

To write a large number in words, split it into groups of three digits from the right. Name each group: thousand, million, billion. 10 007 is ten thousand and seven. Six billion is 6 000 000 000.

Worked example 11 mark

Write the number 707 700 in words.

  1. 1

    split into groups of three: 707 | 700

  2. 2

    seven hundred and seven thousand, seven hundred [1]

2. Prime numbers

A prime number has exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11, 13. The number 1 is not prime. The number 2 is the only even prime.

To test a number, look for a factor other than 1 and itself. 39 is not prime because 3 is a factor of 39. Try the small primes 2, 3, 5, 7 in turn.

Examiner insight

“The most common incorrect choice was 39 although quite a few clearly did not know what prime numbers were by choosing 4 and 22 or the two square numbers, 4 and 25.”

Worked example 12 marks

Write down the two prime numbers that are greater than 20 but less than 30.

  1. 1

    the odd numbers are 21, 23, 25, 27, 29

  2. 2

    21 = 3 × 7, 25 = 5 × 5, 27 = 3 × 9, so none of these is prime

  3. 3

    23 is prime [1]

  4. 4

    29 is prime [1]

3. Square numbers and cube numbers

A square number is a whole number multiplied by itself, such as 5 × 5 = 25. The first are 1, 4, 9, 16, 25, 36. A cube number is a whole number written three times and multiplied, such as 2 × 2 × 2 = 8. The first are 1, 8, 27, 64, 125.

The small raised number is the index. It counts how many times the number is written, so 2³ = 2 × 2 × 2 = 8, not 6.

The difference of two numbers is the larger take away the smaller.

n² = n × n

n³ = n × n × n

Examiner insight

“Those not understanding indices usually gave responses of 10 and 6 leading to an answer of 4.” “A few candidates thought 9 was a cube number.”

Worked example 12 marks

Work out the difference between 5² and 2³.

  1. 1

    5² = 25 and 2³ = 8 [1]

  2. 2

    25 - 8 = 17 [1]

4. Factors, multiples and common factors

A factor of a number divides into it exactly. 1 and the number itself are always factors. Find factors in pairs: for 26, 1 × 26 and 2 × 13.

A multiple of a number is in its times table. Multiples of 9 are 9, 18, 27, 36, 45, 54.

A common factor of two numbers is a factor of both. A common multiple is a multiple of both.

Examiner insight

“Listing the factors of 18 was well done but missing out 1 or 18 was quite common.”

Worked example 13 marks

(a) Write down the multiple of 9 between 50 and 60.

(b) Write down all the common factors of 20 and 30.

  1. 1

    (a) 9 × 6 = 54 [1]

  2. 2

    (b) factors of 20: 1, 2, 4, 5, 10, 20

  3. 3

    factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

  4. 4

    common factors: 1, 2, 5, 10 [2]

5. Prime factors

A whole number above 1 that is not prime can be written as a product of prime factors. Write the primes in order and use a power for repeats. For example, 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3².

Use repeated division. Divide by 2 as often as you can, then by 3, then 5. Divide only by primes. Check by multiplying back.

72 = 2³ × 3²

Examiner insight

“Submitting the three prime factors as the final answer was quite common, even from some who had fully correct working.” “The repeated division method included division by non-primes (9 was common) by quite a number of candidates.”

Worked example 13 marks

(a) Write 84 as a product of its prime factors.

(b) 96 can be written as 2^(P) × 3. Find the value of P.

  1. 1

    (a) 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7 [1]

  2. 2

    84 = 2² × 3 × 7 [1]

  3. 3

    (b) 96 = 2 × 2 × 2 × 2 × 2 × 3, so P = 5 [1]

6. HCF and LCM

The highest common factor (HCF) is the largest number that is a factor of both. The lowest common multiple (LCM) is the smallest number that is a multiple of both.

With prime factors, the HCF takes the lowest power of each prime both numbers share. The LCM takes the highest power of every prime that appears.

The HCF is never bigger than either number. The LCM is never smaller.

Examiner insight

“The most common misunderstanding was to give the HCF, 3.” “It was quite common to lose a mark by giving a multiple of the LCM.”

Worked example 12 marks

Find the lowest common multiple (LCM) of 63 and 42.

  1. 1

    63 = 3² × 7 and 42 = 2 × 3 × 7

  2. 2

    LCM = 2 × 3² × 7 (highest power of each prime) [1]

  3. 3

    = 126 [1]

7. LCM in a timetable problem

When two things repeat every a and b minutes, they happen together after the LCM of a and b minutes. Add that time to the start time.

Work in minutes, then change to hours and minutes. Do not add the two gaps together.

Examiner insight

“A few added 30 and 34 mins to 0800 to give 0904.”

Worked example 13 marks

A red light flashes every 20 minutes. A blue light flashes every 35 minutes. The two lights flash together at 10 00. Find the next time that the two lights flash together.

  1. 1

    20 = 2² × 5 and 35 = 5 × 7

  2. 2

    LCM = 2² × 5 × 7 = 140 minutes [1]

  3. 3

    140 minutes = 2 hours 20 minutes [1]

  4. 4

    10 00 + 2 h 20 min = 12 20 [1]

8. Rational, irrational and reciprocals

A rational number can be written as a fraction of two integers. An irrational number cannot. π and the square root of a number that is not a square, such as √17, are irrational. Terminating decimals such as 4.5 are rational.

The reciprocal of x is 1/x, so the reciprocal of 4 is 1/4. To find the reciprocal of a fraction, turn it upside down: the reciprocal of 5/7 is 7/5.

Examiner insight

“Many responses were terminating decimals between 4 and 5, in particular 4.5”

Worked example 12 marks

(b) Write down the reciprocal of 4.

(c) Write down an irrational number that is between 4 and 5.

  1. 1

    (b) reciprocal of 4 = 1/4 or 0.25 [1]

  2. 2

    (c) 16 < 17 < 25, so 4 < √17 < 5

  3. 3

    17 is not a square number, so √17 is irrational [1]

Recap

  • Natural numbers are the counting numbers. Integers can be positive, zero or negative. 1 is not prime.
  • A prime number has exactly two factors. 2 is the only even prime.
  • Square numbers are n² and cube numbers are n³. A factor divides exactly. A multiple is in the times table.
  • Prime factor form uses powers: 72 = 2³ × 3².
  • HCF uses the lowest powers shared. LCM uses the highest powers of every prime.
  • Irrational numbers are not fractions of integers. The reciprocal of x is 1/x.

End-of-chapter exercise

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

  1. Do not use a calculator. Write the number 40 305 in words.1 mark
  2. Write down all the common factors of 24 and 36.2 marks
  3. Do not use a calculator. Write down the prime number in this list. 51 57 63 67 691 mark
  4. Write down the number in this list that is both a square number and a cube number. 16 25 27 36 64 811 mark
  5. Do not use a calculator. Write 180 as a product of its prime factors.2 marks
  6. Do not use a calculator. Write down the reciprocal of 0.4 as a fraction in its simplest form.2 marks
  7. From this list, write down the irrational numbers. √16 √18 22/7 π √(0.25)2 marks
  8. Do not use a calculator. 360 = 2³ × 3² × 5 and k = 2² × 3⁴ × 7. Find the HCF and the LCM of 360 and k. Leave each answer as a product of prime factors.2 marks
  9. Buses for the airport leave a station every 42 minutes. Buses for the harbour leave the same station every 56 minutes. A bus for each place leaves at 06 45. No bus leaves after 23 00. Find how many more times that day the two buses leave together, and the time of the last of these.5 marks
  10. Do not use a calculator. The HCF of 72 and a whole number n is 12. The LCM of 72 and n is 360. (a) Write 72, 12 and 360 as products of their prime factors. (b) Find the value of n. (c) Find the smallest whole number that n must be multiplied by to give a square number.6 marks

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