Factors, multiples and primes

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교육자를 위해: Factors, multiples and primes(KS3 Maths, Number)을(를) 위한 바로 쓸 수 있는 수업 슬라이드, 복습 노트 — 수업에 사용하거나, 학습자들이 실시간 게임으로 즐기는 인터랙티브 클래스 활동으로 진행하세요.

수업 노트

Factors and Multiples

  • A factor of a number divides it exactly, leaving no remainder.
  • Every number has at least two factors: 1 and itself.
  • A multiple of a number is found by multiplying it by a whole number (1, 2, 3, ...).
  • Every number is a multiple of 1 and of itself.
  • The factors of a number are finite, but its multiples are infinite.
  • Example: the factors of 12 are 1, 2, 3, 4, 6, 12; the first multiples of 12 are 12, 24, 36, 48, ...

Prime Numbers

  • A prime number is a natural number greater than 1 with exactly two positive divisors: 1 and itself.
  • A natural number greater than 1 that is not prime is called a composite number.
  • 1 is not prime because it has only one divisor (itself).
  • The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
  • 2 is the only even prime; every other prime is odd.
  • Any prime greater than 5 ends in 1, 3, 7 or 9.
  • There are infinitely many primes.

Square and Cube Numbers

  • A square number is the result of multiplying a whole number by itself: n × n = n2.
  • The first square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
  • A cube number is the result of multiplying a whole number by itself twice: n × n × n = n3.
  • The first cube numbers are 1, 8, 27, 64, 125, 216.
  • Square numbers have an odd number of factors; all other numbers have an even number of factors.

Prime Factorisation

  • Prime factorisation writes a number as a product of its prime factors.
  • The fundamental theorem of arithmetic says every integer greater than 1 is either prime or can be written as a unique product of primes (apart from order).
  • Use a factor tree: split the number into two factors, then split again until every branch ends in a prime.
  • Use repeated division: divide by the smallest prime that goes in exactly, and repeat until you reach 1.
  • Write the result using index notation, e.g. 360 = 23 × 32 × 5.
  • Example: 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5.

Highest Common Factor (HCF)

  • The highest common factor (HCF) of two numbers is the largest number that divides both exactly.
  • Method 1: list all factors of each number and pick the largest shared one.
  • Method 2: write each number as a product of primes, then multiply the common prime factors (using the lowest power of each).
  • Example: 24 = 23 × 3 and 36 = 22 × 32, so HCF = 22 × 3 = 12.
  • The HCF is never larger than the smaller of the two numbers.

Lowest Common Multiple (LCM)

  • The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple of both.
  • Method 1: list multiples of each number and pick the first shared one.
  • Method 2: write each number as a product of primes, then multiply the highest power of every prime that appears.
  • Example: 24 = 23 × 3 and 36 = 22 × 32, so LCM = 23 × 32 = 72.
  • The LCM is never smaller than the larger of the two numbers.

Venn Diagrams for HCF and LCM

  • Draw two overlapping circles, one for each number.
  • Put the shared prime factors in the overlap.
  • Put the remaining prime factors in the outer parts of each circle.
  • HCF = product of the primes in the overlap.
  • LCM = product of all the primes in the whole diagram.

Solving Word Problems with HCF and LCM

  • Use the HCF when splitting things into equal groups or finding the largest equal-sized piece.
  • Use the LCM when events repeat together or when finding when two things next coincide.
  • Look for key words: 'largest', 'greatest', 'maximum' often signal HCF; 'smallest', 'least', 'next together' often signal LCM.
  • Example (HCF): 24 pencils and 36 pens are shared equally into the greatest number of packs — 12 packs.
  • Example (LCM): one bus comes every 12 minutes and another every 18 minutes; they next leave together after 36 minutes.

Testing for Primality

  • Trial division checks whether a number n is divisible by any prime up to √(n).
  • If no prime up to √(n) divides n, then n is prime.
  • Example: to test 97, check primes up to √(97) ≈ 9.8: 2, 3, 5 and 7. None divide 97, so 97 is prime.
  • Primes are used in public-key cryptography, which relies on how hard it is to factor large numbers into primes.

슬라이드

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연습 문제

무료 미리 보기 — 64개 중 8개 문제. 가입하면 전부 볼 수 있어요.

  1. 1.Which of these numbers is prime?

    Easy
    • A1
    • B2
    • C4
    • D9

  2. 2.Which of these numbers is composite?

    Easy
    • A7
    • B11
    • C15
    • D17

  3. 3.Which of these numbers is a square number?

    Medium
    • A8
    • B9
    • C12
    • D18

  4. 4.Which of these numbers is a cube number?

    Medium
    • A6
    • B8
    • C16
    • D24

  5. 5.Select all the prime numbers in this list. (select all that apply)

    Medium
    • A2
    • B9
    • C13
    • D21
    • E27

  6. 6.1 is a prime number.

    Easy

    True or false?

  7. 7.Every prime number greater than 2 is odd.

    Easy

    True or false?

  8. 8.Match each number to its correct description.

    Easy
    • 2
    • 4
    • 1
    • 9
    • the only even prime number
    • a composite number
    • neither prime nor composite
    • a square number

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