Factors and Multiples: Chapter Review

Vocabulary and notation

Divisible, factor, divisor, multiple

For $d \neq 0$, $n$ is divisible by $d$ when $n \div d$ leaves remainder $0$. Said three ways: $d$ is a factor (or divisor) of $n$, and $n$ is a multiple of $d$.

Digit sum

The total of all the digits, folded again if still awkward to read: $9{,}996 \to 33 \to 6$.

Even, odd

Even means the last digit is $0, 2, 4, 6,$ or $8$, the same property as divisible by $2$. Every other whole number is odd.

Prime

A whole number greater than $1$ with exactly two distinct factors, $1$ and itself. The list starts $2, 3, 5, 7, 11, 13$; $2$ is the only even entry.

Composite

A whole number greater than $1$ with more than two distinct factors. Every whole number greater than $1$ is exactly one of prime or composite; $1$ is neither, having only the factor $1$.

Twin primes

Two primes differing by $2$: $11$ and $13$.

Prime factorization

A whole number greater than $1$ written as a product of primes only: $84 = 2 \times 2 \times 3 \times 7$. A number already prime is its own factorization.

Prime-power form

The factorization with equal primes collected into powers, smallest prime first: $84 = 2^2 \times 3 \times 7$. An exponent of $1$ is left unwritten.

Common factor, $\operatorname{GCF}(a, b, \ldots)$

A common factor divides all the numbers; the largest is the greatest common factor. Any list shares $1$, so a GCF always exists.

Common multiple, $\operatorname{LCM}(a, b, \ldots)$

A multiple of all the numbers; the smallest other than $0$ is the least common multiple. Common multiples never run out.

Formulas and theorems

• The divisibility tests

  $$\begin{array}{r|l} 2 & \text{last digit } 0, 2, 4, 6, 8 \\ 3 & \text{digit sum a multiple of } 3 \\ 4 & \text{last two digits a multiple of } 4 \\ 5 & \text{last digit } 0 \text{ or } 5 \\ 6 & \text{passes the } 2 \text{ and } 3 \text{ tests} \\ 8 & \text{last three digits a multiple of } 8 \\ 9 & \text{digit sum a multiple of } 9 \\ 10 & \text{last digit } 0 \end{array}$$

  Use when Whole numbers in base ten. These are the digit tests; for $7$ or $11$, divide directly.

• Multiples of $d$ add and subtract

  $$(d \times m) \pm (d \times n) = d \times (m \pm n)$$

  Use when Whole numbers, with $m \ge n$ so the difference stays whole. A sum or difference of two multiples of $d$ is again one, so split off a part plainly divisible by $d$ and test only the leftover.

  e.g. $714 = 700 + 14$, and $7$ divides both pieces, so $7$ divides $714$.

• A factor of a factor is a factor

  $$\begin{gathered} e \text{ divides } d \text{ and } d \text{ divides } n\\ \Longrightarrow\ e \text{ divides } n \end{gathered}$$

  Use when Whole numbers, one direction only. Divisible by $9$ forces divisible by $3$, and by $10$ forces divisible by $2$ and $5$; divisible by $3$ forces nothing about $9$.

• Remainder from the digit sum

  $$\begin{gathered} n \text{ and its digit sum leave}\\ \text{the same remainder on}\\ \text{division by } 9, \text{ and by } 3 \end{gathered}$$

  Use when Base ten whole numbers. Fold as often as needed; a fold landing on $9$ means remainder $0$, not $9$.

  e.g. $4{,}271$ has digit sum $14$, which leaves $5$ on division by $9$, so $4{,}271$ leaves $5$.

• Combined test for a composite divisor

  $$\begin{gathered} n \text{ is divisible by } a \times b\\ \iff n \text{ is divisible by } a \text{ and by } b \end{gathered}$$

  Use when Requires $\operatorname{GCF}(a, b) = 1$. Workable splits: $12 = 4 \times 3$, $18 = 2 \times 9$, $20 = 4 \times 5$, $24 = 8 \times 3$, $36 = 4 \times 9$, $45 = 9 \times 5$, $72 = 8 \times 9$. Left to right holds unconditionally; only right to left needs the check.

• Stopping rule for trial division

  $$\begin{gathered} \text{test } d = 2, 3, 5, 7, 11, \ldots\\ \text{only while } d \times d \le n \end{gathered}$$

  Use when $n$ greater than $1$. The bound is not strict: at $n = 49$ you still test $d = 7$. Any divisor found makes $n$ composite; none found in range makes it prime.

  e.g. For $127$ the largest $d$ with $d \times d \le 127$ is $11$, so $2, 3, 5, 7, 11$ settle it.

• Fundamental Theorem of Arithmetic

  $$\begin{gathered} n = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}\\ \text{unique apart from order} \end{gathered}$$

  Use when Every whole number greater than $1$ has exactly one, with $p_1 < p_2 < \cdots < p_k$ prime and every $a_i \ge 1$. A factor tree's starting split never changes it, and $1$, not being prime, never appears.

  e.g. $8 \times 9$ and $6 \times 12$ both finish at $72 = 2^3 \times 3^2$.

• Reading factors off a factorization

  $$\begin{gathered} d \text{ is a factor of } n\\ \iff d \text{ uses only primes of } n,\\ \text{each to at most its exponent in } n \end{gathered}$$

  Use when $n$ greater than $1$ in prime-power form; $d = 1$ qualifies, using no primes. A prime absent from the factorization can never divide $n$.

  e.g. $210 = 2 \times 3 \times 5 \times 7$: $35 = 5 \times 7$ is a factor, $11$ is not.

• Perfect squares and cubes from the exponents

  $$\begin{gathered} n \text{ is a perfect square}\\ \iff \text{every exponent is even}\\ n \text{ is a perfect cube}\\ \iff \text{every exponent is a multiple of } 3 \end{gathered}$$

  Use when $n$ greater than $1$ in prime-power form, and every exponent must qualify, not just one.

  e.g. $784 = 2^4 \times 7^2 = 28^2$ is a square; $2^4 \times 7 = 112$ is not.

• GCF and LCM from prime factorizations

  $$\begin{gathered} \operatorname{GCF}: \text{lowest power}\\ \text{of each shared prime}\\ \operatorname{LCM}: \text{highest power}\\ \text{of every prime that appears} \end{gathered}$$

  Use when Whole numbers greater than $1$ in prime-power form. A prime present in only some adds nothing to the GCF and its full power to the LCM. Both rules extend to three or more numbers.

• GCF times LCM

  $$\begin{gathered} \operatorname{GCF}(a, b) \times \operatorname{LCM}(a, b) = a \times b\\ \operatorname{LCM}(a, b) = (a \times b) \div \operatorname{GCF}(a, b) \end{gathered}$$

  Use when Exactly two positive whole numbers, and no rule of this shape for three: $\operatorname{GCF}(2, 3, 4) \times \operatorname{LCM}(2, 3, 4) = 12$, not $24$.

  e.g. $18$ and $30$: $6 \times 90 = 540$, and $18 \times 30 = 540$.

• Size check on a GCF or an LCM

  $$\operatorname{GCF}(a, b) \le a, b \le \operatorname{LCM}(a, b)$$

  Use when Positive whole numbers. Both become equalities exactly when the smaller divides the larger: $7$ and $14$ give GCF $7$, LCM $14$. And for two numbers, $\operatorname{GCF}(a, b) = 1$ exactly when $\operatorname{LCM}(a, b) = a \times b$.

Problem types, step by step

Exam traps

• Trap Reading a digit sum that is a multiple of $3$ as proof of divisibility by $9$.

  Fix Only a digit sum that is a multiple of $9$ clears $9$. The digit sum of $888$ is $24$, a multiple of $3$ but not $9$: $888 \div 3 = 296$, but $888 \div 9$ leaves $6$.

• Trap Splitting a composite divisor into factors that share a factor, testing $12$ as "$2$ and $6$" or $4$ as "$2$, twice".

  Fix The split needs $\operatorname{GCF} = 1$. $18$ passes the $2$ and $6$ tests yet $18 \div 12$ is not whole; $12 = 4 \times 3$ correctly fails it.

• Trap Quitting trial division after the first few primes fail.

  Fix Keep going while $d \times d \le n$. Stopping on $391$ after $13$ misses $391 = 17 \times 23$, since $19 \times 19 = 361 < 391$.

• Trap Treating $1$ as prime, or assuming every prime is odd and every odd number prime.

  Fix $1$ has a single factor, so it is neither. $2$ is prime and even, while $9$, $15$, $21$, $27$ are odd and composite.

• Trap Handing in a partial split as a prime factorization, as $24 = 2^2 \times 6$.

  Fix Every factor written must be prime. Split the $6$ and finish at $24 = 2^3 \times 3$.

• Trap Dropping an unshared prime from the LCM, or swapping the lowest-power and highest-power rules.

  Fix A prime in only one number joins the LCM at full power. For $8 = 2^3$ and $9 = 3^2$: $\operatorname{LCM} = 72$, $\operatorname{GCF} = 1$, never $\operatorname{LCM} = 1$.

• Trap Handing in $a \times b$ as the LCM.

  Fix That is a common multiple, least only when $\operatorname{GCF}(a, b) = 1$. For $6$ and $10$ the product $60$ is not the LCM $30$; divide the product by the GCF.

• Trap Carrying $\operatorname{GCF} \times \operatorname{LCM} = a \times b$ over to three numbers.

  Fix Guaranteed for a pair, no rule for three: $\operatorname{GCF}(2, 3, 4) = 1$ and $\operatorname{LCM}(2, 3, 4) = 12$ multiply to $12$, not $24$. Use the highest powers for a three-number LCM.