Number types and prime factors

Explanation

Conventions and the real-number family

In this pack, natural numbers mean the counting numbers 1, 2, 3, … . Some books include 0, so a question should state its convention when that difference matters. Here, 0 is an integer but not a natural number.

The number sets are nested. Every natural number is an integer, every integer is rational, and every rational number is real. Irrational numbers are also real, but they are not rational. A number may therefore have more than one correct classification; use the most specific requested description.

Explanation

Natural numbers, integers and zero

Natural numbers are positive counting numbers. Integers are whole numbers and their negatives, including zero: …, −3, −2, −1, 0, 1, 2, 3, … . Integers do not contain fractional or decimal parts.

Zero needs careful language. It is an integer, a rational number and a real number. It is neither positive nor negative. Under this pack's convention it is not natural.

Explanation

Rational, irrational and real numbers

A rational number can be written as a/b, where a and b are integers and b ≠ 0. Integers, fractions, terminating decimals and recurring decimals are rational. For example, −4 = −4/1, 0 = 0/1 and 0.125 = 1/8.

An irrational number cannot be written as a ratio of two integers. Its decimal expansion neither terminates nor repeats. Examples include √2, √5 and π. A root is not automatically irrational: √49 = 7 is rational. Rational and irrational numbers together form the real numbers used on the number line.

Explanation

Prime numbers

A prime number is an integer greater than 1 with exactly two positive factors: 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13. The number 2 is the only even prime.

The number 1 is not prime because it has only one positive factor. A composite number is an integer greater than 1 with more than two positive factors. Zero and negative integers are neither prime nor composite in this course.

Explanation

Square and cube numbers

A square number has the form n² for an integer n. The non-negative square numbers begin 0, 1, 4, 9, 16, 25, 36, 49 and 64. Although (−5)² and 5² both equal 25, no negative real number is a square of a real integer.

A cube number has the form n³. Cubes may be negative, zero or positive: (−3)³ = −27, 0³ = 0 and 4³ = 64. Recognising squares and cubes helps with classification and factor problems.

Explanation

Factors, multiples and divisibility

A factor divides a number exactly, leaving no remainder. For example, the positive factors of 18 are 1, 2, 3, 6, 9 and 18. Factors of a positive integer form a finite list.

A multiple is produced by multiplying a number by an integer. Positive multiples of 6 include 6, 12, 18, 24, … . Multiples continue without end. If 6 is a factor of 18, then 18 is a multiple of 6.

Explanation

Common factors and common multiples

A common factor divides each of two or more numbers exactly. For 12 and 18, the positive common factors are 1, 2, 3 and 6. Their highest common factor is therefore 6.

A common multiple is a multiple of each number. Common multiples of 4 and 6 include 12, 24 and 36. Their lowest positive common multiple is 12.

Explanation

Prime factorisation

A prime factorisation writes an integer greater than 1 as a product of primes. Divide repeatedly by small primes or use a factor tree, then collect repeated factors with indices. Apart from order, the result is unique.

For example, 180 = 2 × 90 = 2 × 2 × 45 = 2² × 3² × 5. Check by multiplying: 4 × 9 × 5 = 180. The final expression must contain prime factors only.

Explanation

Highest common factor

The highest common factor (HCF) is the greatest positive integer that divides every given number. Listing factors is efficient for small values. Prime factorisation is more reliable for larger values.

Using prime factors, keep only primes shared by all numbers and choose the smallest power of each shared prime. The smaller power is the greatest amount guaranteed to divide every number.

Explanation

Lowest common multiple

The lowest common multiple (LCM) is the smallest positive integer that is a multiple of every given number. Listing multiples works for small values. Prime factors avoid long lists.

Using prime factors, include every prime that appears and choose its greatest power in any one number. This supplies enough of every prime factor for each original number to divide the result.

Explanation

Reciprocals

The reciprocal of a non-zero number x is 1/x. Multiplying a non-zero number by its reciprocal gives 1. For a fraction a/b with a ≠ 0 and b ≠ 0, interchange numerator and denominator: the reciprocal is b/a.

Examples: the reciprocal of 5 is 1/5, of −3/4 is −4/3, and of 0.2 is 5 because 0.2 × 5 = 1. The reciprocal of 0 is undefined: there is no number that gives 1 when multiplied by 0.

Explanation

Choosing between HCF and LCM in context

Use HCF when a fixed total is split into the largest equal groups, pieces or measurements with nothing left over. Clues include greatest size, largest equal group and maximum length.

Use LCM when cycles repeat and you need the first time or smallest quantity at which they coincide. Clues include together again, first common time and smallest number that can be packed in several ways. Always interpret the numerical result in the context.

Revision checklist and error traps

Classification: state the natural-number convention; remember that 0 is rational and real but is not positive; check whether a square root simplifies before calling it irrational; and remember that 1 is not prime.

Factors and multiples: a factor divides, while a multiple is produced by multiplication. For HCF take shared prime factors with smallest powers. For LCM take all required prime factors with greatest powers.

Reciprocals and context: never take the reciprocal of 0. In word problems, explain why HCF or LCM fits, calculate, then attach the correct unit or meaning to the result.