ACARA v9 CONTENT DESCRIPTION “express natural numbers as products of their factors, recognise multiples and determine if one number is divisible by another”
Builds on the multiplication and division facts of earlier years, and on seeing those two operations as partners. Year 5 now looks inside whole numbers: every counting number can be built from smaller numbers multiplied together, it sits in a steady rhythm among its multiples, and it either divides evenly by another number or leaves a remainder. Factors, multiples and divisibility are three views of the same multiplicative world, and together they prepare the way for fractions, primes and much of the number work to come.
Factors are the building blocks
A factor of a number is a whole number that divides into it exactly, with nothing left over. Twelve has the factors one, two, three, four, six and twelve, because each of them divides twelve evenly. Factors are the building blocks of a number under multiplication: pick two factors that multiply to the number and you have taken it apart. The simplest picture is a rectangle, because a number of dots can be arranged into a neat rectangle only when its sides are factors. A child who looks for rectangles is really hunting for factors, and the search has a tidy, finite end.
A factor pair is a rectangle
A number fits a rectangle only when its sides are factors.
1 × 12 makes 12, so 1 and 12 are a factor pair of 12.
Factors come in pairs
Factors never arrive alone; they come in pairs that multiply to the number. For twelve, one pairs with twelve, two with six, and three with four, so the six factors are really three partnerships. This is why a rectangle works both ways: three rows of four is the same twelve as four rows of three. Listing factors in pairs makes sure none is missed, and it shows when a number is special, because a square number has one factor that pairs with itself. Seeing the pairing turns factor-finding from random guessing into an orderly sweep inward from one and the number itself.
Factors come in pairs
List factors in pairs so none is missed.
Each factor of 24 has a partner it multiplies with to make 24. Reveal each partner.
Multiples run the other way
Where factors look inward, multiples look outward. The multiples of a number are what you reach by counting in steps of it: the multiples of six are six, twelve, eighteen, twenty-four and onward, with no end. Factors and multiples are two ends of the same relationship, because if four is a factor of twenty-four, then twenty-four is a multiple of four. Multiples lay down a steady rhythm along the number line, and recognising that rhythm is the heart of skip counting, of times tables, and of spotting when one number will appear in another's count.
Multiples run the other way
Count in steps of a number to reach its multiples.
Counting in 3s lands on the multiples of 3: 3, 6, 9, and on without end.
Divisible means no remainder
To say one number is divisible by another is just to say the division comes out exactly, with a remainder of zero. Twenty-four is divisible by six because twenty-four shared into groups of six leaves none over; thirty is not divisible by four, because four groups of seven reach twenty-eight and two are left stranded. Divisibility is the test that decides whether one number is a factor of another, and the remainder is the evidence. Checking it carefully, rather than guessing, is what tells a child whether a sharing will be fair or whether something will be left behind.
Divisible means no remainder
One number is divisible by another only if it divides exactly.
Is 24 divisible by 6? Divisible means it divides with no remainder.
Writing a number as a product
Because every number is built from factors, it can be written as a product of them. Thirty-six is four times nine, and also six times six, and either is an honest factoring. Writing a number as a product of its factors is the first step toward the idea of prime numbers, those that can only be written as one times themselves, and toward breaking a number all the way down into its prime building blocks. For now, the skill is simply to choose a factor pair and write the number as their product, which keeps the multiplicative structure of the number in plain view.
Writing a number as a product
Every number can be written as a product of its factors.
What two factors multiply to make 36?
Factors, multiples and divisibility together
Factors, multiples and divisibility are one idea seen from three sides. A factor divides a number exactly; a multiple is reached by counting in steps of a number; and divisibility is the test that links them, since one number is a factor of another precisely when it divides with no remainder. Holding the three together lets a child move fluently between them: from a factor pair to a product, from a step size to its multiples, from a division to a yes-or-no about remainders. This multiplicative fluency is what the fractions, primes and problem solving of the rest of Year 5 will lean on.