Factors, Multiples, Primes and Special Numbers
Every number has factors
A factor of a number is a whole number that divides it exactly, with nothing left over. The number 12 has factors 1, 2, 3, 4, 6 and 12, because each one fits into 12 a whole number of times. A neat way to see factors is as rectangles: 12 squares can be arranged as 1 by 12, 2 by 6 or 3 by 4, and each rectangle that fills completely shows a factor pair. Every number can be built this way, and reading off its factors is the first step to understanding how numbers are put together.
Multiples count in steps
Where factors look inside a number, multiples look outward. The multiples of a number are what you reach by counting up in equal steps: the multiples of 3 are 3, 6, 9, 12 and on, and the multiples of 5 are 5, 10, 15, 20. Every number has endlessly many multiples, one for each step you take. Factors and multiples are two sides of the same relationship: 3 is a factor of 12, and 12 is a multiple of 3. Holding both ideas together is what makes the rest of this unit work.
Primes and composites
Some numbers can be made into only one rectangle, a single row. A prime number has exactly two factors, 1 and itself, so 7, 13 and 23 are prime: they cannot be split into smaller equal groups. A composite number has more than two factors, like 12 or 15, so it can be arranged as several different rectangles. The number 1 is neither, having just one factor. Sorting numbers into prime and composite reveals which are the indivisible building blocks and which are built from them.
Factors two numbers share
When two numbers are compared, the factors they have in common matter. Both 12 and 18 have 1, 2, 3 and 6 among their factors, so these are their common factors, and the largest of them, 6, is the highest common factor. Finding the highest common factor is what lets a fraction be simplified to its lowest terms, or a quantity be shared into the largest equal groups. It is one of the most useful things factors are good for, turning two separate lists into a single shared answer.
Where multiples meet
Multiples lead to a matching idea. Counting in fours gives 4, 8, 12, 16; counting in sixes gives 6, 12, 18; the first number that appears in both is 12, the lowest common multiple. The lowest common multiple is what you need to add fractions with different denominators, or to work out when two repeating events line up again. Just as the highest common factor is the largest factor two numbers share, the lowest common multiple is the smallest multiple they share, and the two ideas work as a pair.
Breaking down into primes
Every composite number can be broken down into primes, and in only one way. Splitting 12 into 2 times 6, then 6 into 2 times 3, leaves 2, 2 and 3, all prime: this is the prime factorisation of 12. A factor tree is a tidy way to carry out the splitting, branching each number into factors until only primes remain at the tips. Because this breakdown is unique to each number, prime factorisation is a kind of fingerprint, and it underlies how highest common factors and lowest common multiples can be found quickly.
Square and triangular numbers
Some numbers are named for the shape they build. A square number is a whole number multiplied by itself: 1, 4, 9, 16 and 25 each fill a perfect square, because 3 times 3 is 9 and 4 times 4 is 16. Square numbers are also the only numbers with an odd count of factors, because the square root pairs with itself instead of with a partner. A triangular number is the running total of the counting numbers: 1, then 1 plus 2 is 3, then plus 3 is 6, then plus 4 is 10, stacking into a neat triangle of dots. Spotting these patterns lets you predict the next number in the sequence and explain why it grows the way it does, the same kind of hidden structure that factors and primes reveal.
Why this structure matters
Factors, multiples and primes are the grammar of whole numbers. Knowing them lets you simplify fractions, find common denominators, solve sharing and grouping problems, and recognise the patterns that square and triangular numbers follow. These ideas reach forward into every later topic that uses number: ratio, algebra and beyond all rest on being able to see a number not just as a quantity, but as something with an inner structure of factors and primes waiting to be used.