C1.1; E1.1 - Factors, multiples and primes

Factor pairs

You have $24$ square tiles. How many different rectangles can you make using every tile, if turning a rectangle does not count as a new arrangement? The possible side lengths reveal a useful structure inside the number $24$.

For example, $3$ rows of $8$ tiles use exactly $24$ tiles because $3\times8=24$. A row length of $5$ leaves tiles over, so $5$ is not a factor of $24$.

[DIAGRAM: asset_name: Rectangles from factor pairs; asset_slug: factor_pair_rectangles; caption: Twenty-four tiles make four rectangles, counting rotations as the same arrangement.; description: Four exact tile arrays labelled one by twenty-four, two by twelve, three by eight and four by six. Each array contains twenty-four equal squares. The caption identifies the corresponding factor pairs.]

Test possible divisors in order, starting at $1$. Every successful division gives two factors at once. Once the smaller member of the next pair would be greater than the larger member, you have already found the pairs in the opposite order. You can stop without testing every number up to $24$.

Change the number to $25$. This time a pair has equal members, which changes how many distinct factors it contributes.

A number such as $25=5\times5$ is a square number: a whole number multiplied by itself. The middle factor appears only once in the factor list, even though it appears twice in its pair.

Prime and composite numbers

A row of $13$ tiles can only form the rectangle $1\times13$. In contrast, $15$ tiles can form $1\times15$ or $3\times5$. The distinction depends on the number of factors, not on how large or awkward the number looks.

The number $1$ has only one positive factor, $1$, so it is neither prime nor composite. Zero and negative integers are also outside these definitions.

The number $2$ has precisely the factors $1$ and $2$, so it is prime. Every other positive even integer is greater than $2$ and has $2$ as an extra factor. Therefore $2$ is the only even prime. Being odd does not guarantee that a number is prime: $9=3\times3$ and $21=3\times7$ are odd composite numbers.

To decide whether $29$ is prime, test possible small factors. Neither $2,3,4$ nor $5$ divides it exactly. A pair with both members at least $6$ would have product at least $36$, too large. Any remaining factor pair would therefore have a smaller partner that has already been tested. The only pair is $1\times29$, so $29$ is prime.

Here are the primes below $20$, found using the same definition:

Multiples and divisibility

Factors look inside a number; multiples are made by taking whole groups of a number. The same multiplication links both ideas: because $6\times7=42$, $6$ and $7$ are factors of $42$, and $42$ is a multiple of each.

Zero is a multiple of every integer, because multiplying by $0$ gives $0$. When asked for the “lowest common multiple” later, you will look for a positive multiple. For a positive number, there are finitely many positive factors, but its positive multiples continue without end.

Useful divisibility checks follow from place value. A whole number is divisible by $2$ if its units digit is $0,2,4,6$ or $8$; by $5$ if its units digit is $0$ or $5$; and by $10$ if its units digit is $0$. Each ten is already divisible by $2$ and by $5$, so the units decide the remainder.

A whole number is divisible by $3$ if its digit sum is divisible by $3$, and similarly for $9$. Why? Each ten is $9+1$, each hundred is $99+1$, and each higher place value is a multiple of $9$ plus $1$. Replacing place values by $1$ therefore leaves the same remainder on division by $9$, and also by $3$. For example, $243=2\times99+4\times9+(2+4+3)$; the digit sum is $9$, so $243$ is divisible by $9$.

Increase the number by $1$, keeping the proposed factor unchanged. The remainder now decides the answer.

Return to the tiles: the four factor pairs of $24$ give four rectangles. Its eight distinct factors do not give eight different rectangles, because turning $3\times8$ into $8\times3$ keeps the same arrangement.

Quick Recap

• Factors divide a positive integer exactly; factor pairs multiply to it.
• A prime has exactly two distinct positive factors. The number $1$ is neither prime nor composite, and $2$ is the only even prime.
• A multiple is made by whole-number groups; a remainder means the division is not exact.
• “Factor of” and “multiple of” describe opposite directions of the same multiplication relationship.

The $24$ tiles make four different rectangles: $1\times24$, $2\times12$, $3\times8$ and $4\times6$, counting turns as the same arrangement.