Highest Common Factor and Lowest Common Multiple: Grade 9 Math

A periodical cicada with a black body, red eyes and clear wings with orange veins, clinging to a white-painted wooden wall — **Figure 10.1** A periodical cicada of Brood XIII, the brood that comes out every 17 years. It spends those years underground as a nymph. Photo: Joelmills, CC BY-SA 3.0

In the spring of 2024, two broods of periodical cicadas came out of the ground in Illinois, in the United States, in the same season. One brood appears every 17 years, the other every 13. It was the first time the two had come out together since 1803. When will they next?

The first brood is out in the years 2024 + 17, 2024 + 34, …; the second in 2024 + 13, 2024 + 26, …. The next year in both lists is 2024 plus the lowest common multiple (LCM) of 13 and 17. This unit finds the LCM, and the highest common factor (HCF), of expressions with letters, so that one answer holds for every number at once. In Back to the cicadas you will see why these two meet so rarely.

Each tick is a year when that brood comes out; the green lines are the years when both do. Set the two cycles to 13 and 17, then to 12 and 18.

One brood, years apart: 13 The other, years apart: 17

HCF: 1
LCM: 221 years
Together again: 2245

13 and 17 share no factor (HCF 1), so the first year in both lists is 13 × 17 = 221 years on: next in 2245.

For the exam

The question: Grade 9’s final exam follows CDC’s specification grid for Compulsory Mathematics, Grades 9–10, where Algebra is 15 marks in three questions. In the grid’s Grade 9 model paper, question 9a is find the HCF of $x^3 + 27$ and $x^4 + 9x^2 + 81$ (3 marks, understanding): two factorizations, then the HCF.
What it needs:

$a^2 - b^2 = (a + b)(a - b)$ , $a^3 \pm b^3$ and $a^4 + a^2b^2 + b^4 = (a^2 + ab + b^2)(a^2 - ab + b^2)$ from Unit 9.
HCF = the product of the common factors.
LCM = the common factors × the remaining factors.
For two expressions: HCF × LCM = first expression × second expression.

How the exam asks it has a full question marked part by part; Exam practice has every exercise question to mark yourself.

10.0 Review

You met the HCF and LCM of numbers in earlier grades. Take 12 and 18, and write each as a product of primes:
$\begin{aligned} 12 &= 2 \times 2 \times 3 \\ 18 &= 2 \times 3 \times 3 \end{aligned}$

The HCF takes what both have: one 2 and one 3. HCF = 2 × 3 = 6, the biggest number that divides both.
The LCM takes everything either has, each prime as many times as it appears in the one that has it most: two 2s (from 12) and two 3s (from 18). LCM = 2 × 2 × 3 × 3 = 36, the smallest number both divide.

Notice 6 × 36 = 216 = 12 × 18. That is no accident, and it comes back in 10.2.

Letters work the same way, with each letter’s power counting how many times it is multiplied. $6x^2y = 2 \times 3 \times x \times x \times y$ and $9xy^3 = 3 \times 3 \times x \times y \times y \times y$ . Both have one 3, one x and one y: HCF = 3xy. Everything, each the most times: LCM $= 2 \times 3^2 \times x^2 \times y^3 = 18x^2y^3$ . So for a letter, the HCF takes its lowest power and the LCM its highest.

Expressions with more than one term have to be factorized first (Unit 9). Then their factors play the part of the primes.

What is the HCF of $8a^3b^2$ and $12a^2b^4$ ?

Pick an answer to see why.

10.1 Highest Common Factor

To find a common factor, every expression has to be broken into its factors first. The usual order:

Take out any common factor of the terms (a number, a letter).
Look for a formula the rest fits: $a^2 - b^2$ , $a^3 \pm b^3$ , $a^4 + a^2b^2 + b^4$ , a perfect square, or a middle term to split.
Factorize it, and keep going until no bracket splits any further.

The product of the common factors of the given expressions is called their highest common factor (HCF). If a factor appears more than once in each, it goes in the HCF as many times as it appears in the one that has it fewest times.

Why “highest”? Any common factor divides both expressions, and the HCF is the one of highest degree (the biggest power of the letter): 1 and x − 2 both divide $x^2 - 4$ and $x^3 - 8$ , but x − 2 has degree 1 and the number 1 has degree 0. For numbers, “highest” means largest; for expressions, it means highest degree.

Try it: tap a factor of A that B also has.

A factor both expressions have moves into the middle, once. When nothing else is shared, read the HCF and LCM off the picture.

Expressions

Tap a factor of A = x² − 4 that is also a factor of B = x³ − 8.

What is the HCF of $(x + 2)^2$ and $x^2 - 4$ ?

Pick an answer to see why.

Exercise 10.1

Question 1

What is meant by the HCF of algebraic expressions?

Answer

When is the HCF of algebraic expressions 1?

Answer

What is the product of the common factors of $4x^3y$ and $3z^3$ ?

Answer

Question 2

Find the HCF.

$(x + y)^2$ and (x + y)(x − y)

Answer

$(x - y)(x^2 + xy + y^2)$ and $(x^2 + xy + y^2)(x^2 - xy + y^2)$

Answer

$(2x - 3y)(4x^2 + 6xy + 9y^2)$ and 4xy(2x − 3y)(2x + 3y)

Answer

$4a^3b(a - b)(a + b - 1)$ and $16a^2b^2(a + 2b)(a + b - 1)$

Answer

Question 3

Find the HCF.

$a^2 - ab$ and $a^3b - a^2b^2$

Answer

$3x^2 + 9x$ and 7x + 21

Answer

$a^3 + 1$ and $a^4 + a^2 + 1$

Answer

$8x^3 + 27y^3$ and $16x^4 + 36x^2y^2 + 81y^4$

Answer

$m^3 + 1 + 2m^2 + 2m$ and $m^3 - 1$

Answer

$8(6x^4 - x^3 - 2x^2)$ and $12(2x^6 + 3x^5 + x^4)$

Answer

$a^2 + 4ab + 4b^2 - c^2$ and $a^2 - 4b^2 + ac - 2bc$

Answer

$x^4 + 4y^4$ and $2x^3y + 4xy^3 + 4x^2y^2$

Answer

$x^2 - 10x + 24 + 6y - 9y^2$ and $x^2 + 3xy - 6x$

Answer

$(1 - x^2)(1 - y^2) + 4xy$ and $1 - 2x + y - x^2y + x^2$

Hint

Answer

$8a^3 + 1$ and $16a^4 - 4a^2 + 4a - 1$

Answer

$2a^3 - a^2 + a - 2$ and $a^3 - a^2 + a - 1$

Answer

Question 4

Find the HCF.

$a^3 + b^3$ , $a^3 - a^2b + ab^2$ and $a^4 + a^2b^2 + b^4$

Answer

$x^3 + 2x^2 + 2x + 1$ , $x^3 - 1$ and $x^4 + x^2 + 1$

Answer

$x^3 - 4x$ , $4x^3 - 10x^2 + 4x$ and $3x^4 - 8x^3 + 4x^2$

Answer

$y^2 + 2y - 8$ , $y^2 - 5y + 6$ and $y^2 + 5y - 14$

Answer

$x^2 + 2x + 1$ , $x^2 + 5x + 6$ and $2x^2 - 5x + 2$

Answer

$x^2 + 2x - 8$ , $x^2 - 2x - 24$ and $x^2 + 5x + 4$

Answer

$x^3 + 2x^2 + 4x$ , $x^4 + 4x^2 + 16$ and $x^3 - 8$

Answer

$8x^3 + 27y^3$ , $16x^4 + 36x^2y^2 + 81y^4$ and $4x^3 - 6x^2y + 9xy^2$

Answer

$2x^3 - 54$ , $24x^4 + 18x^2 + 162$ and $2x^2 + 6x + 18$

Answer

$9x^2 - 4y^2 - 8yz - 4z^2$ , $4z^2 - 4y^2 - 9x^2 - 12xy$ and $9x^2 + 12xz + 4z^2 - 4y^2$

Answer

$2ax^2 + 2ax - 12a$ , $3a^2x^2 - 7a^2x - 6a^2$ and $a^3x^2 + 4a^3x - 21a^3$

Answer

$x^3 + 64y^3$ , $x^4 + 16x^2y^2 + 256y^4$ and $4x^3 - 16x^2y + 64xy^2$

Answer

$(a + b)^3 - 3ab(a + b)$ , $a^4 + a^2b^2 + b^4$ and $a^4 - 2a^3b + a^2b^2 - b^4$

Answer

Question 5

The areas of three rectangular rooms are (x + 3)(x + 6), $x^2 + 8x + 15$ and $x^2 + 7x + 12$ square units. The rooms all have the same breadth. Find it.

Answer

10.2 Lowest common multiple

The product of the common factors and the remaining factors of the given expressions is called their lowest common multiple (LCM). Each factor goes in as many times as it appears in the expression that has it most often. The LCM is the expression of lowest degree that each of the given expressions divides exactly.

Why is HCF × LCM the product of the two? Look at the Venn diagram. The first expression is its own oval: the middle times its own side. The second is the middle times its side. Their product therefore has the middle twice, and each side once. The HCF is the middle once; the LCM is the middle once and both sides. So HCF × LCM also has the middle twice and each side once. For two expressions:
$\text{HCF} \times \text{LCM} = \text{first} \times \text{second}$

The LCM of $(x - 1)^2$ and $x^2 - 1$ is:

Pick an answer to see why.

$(x - 1)^2 = (x - 1)(x - 1)$ and $x^2 - 1 = (x - 1)(x + 1)$ . The LCM takes each factor the most times it appears in either: (x − 1) twice (from the first) and (x + 1) once: $(x - 1)^2(x + 1)$ . Multiplying the two expressions gives $(x - 1)^3(x + 1)$ , a common multiple but not the lowest. x − 1 is the HCF.

When you find the LCM of three expressions, a factor that only two of them share still counts as a common factor: it goes in once (as many times as the expression with most of it has it), never once for each expression. For the HCF, a factor must be in all of them.

Back to the cicadas

13 and 17 are both prime, so they have no common factor: their HCF is 1. Then HCF × LCM = 13 × 17 gives
$\text{LCM} = \frac{13 \times 17}{1} = 221$
The two broods next come out together in 2024 + 221 = 2245.

What it means: two cycles that share no factor meet only once in the product of their lengths. Cycles of 12 and 18 years share the factor 6, so they would meet every 12 × 18 ÷ 6 = 36 years, about six times as often.

Biologists have suggested that this is one reason these cicadas keep prime cycles: a prime cycle meets a predator’s or another brood’s shorter cycle rarely. A predator that peaks every 4 years would meet a 12-year brood every 12 years, but a 13-year brood only every 4 × 13 = 52 years. (It is a hypothesis; another is that it keeps broods from mixing.)

How the exam asks it

Grade 9’s final exam is set to CDC’s specification grid for Compulsory Mathematics, Grades 9–10. There, Algebra is 15 marks in three questions. In the grid’s Grade 9 model paper, the HCF is question 9a:

Part	Level	Marks	What it asks
a.	understanding	3	the HCF of a sum or difference of cubes and an expression $a^4 + a^2b^2 + b^4$

Its three marks are for three steps, factorizing each expression and then picking out the HCF, so write both factorizations in full even when you can see the answer. Here is one in that shape.

Exercise 10.2

Question 1

Define LCM.

Answer

For two expressions, how are their HCF and LCM related to the expressions?

Answer

Question 2

Find the LCM.

2x(x + 2)(x − 2) and $4x^2(3x + 7)(x - 2)$

Answer

$3x^2y(x - y)(x^2 + 3xy + 9y^2)$ and $10x(x^2 + y^2)(x + y)(x - y)$

Answer

$(2x - y)(x^2 + xy + y^2)(x^2 - xy + y^2)$ and 25xy(2x − y)

Answer

$8x^3y^2(a + b + 1)(a - b + 1)$ and $5x^3y(a + b + 1)(a + b + 2)$

Answer

Question 3

The HCF and LCM of two expressions are $(a + b)^2$ and $3a(a + b)^2(2a + b)$ , and the second expression is $3a(a + b)^2$ . Find the first expression.
The textbook gives the LCM as $3a^2(a + b)(2a + b)$ , but its numbers don’t work: an LCM always contains the HCF and each expression, and that one has (a + b) only once. With the LCM $3a(a + b)^2(2a + b)$ they fit.

Answer

Question 4

Two expressions are (x + 5) and $(x + 5)(x^2 - 5x + 25)$ , and their HCF is (x + 5). Find their LCM.

Answer

Question 5

Find the LCM.

$x^2 - x + 1$ and $x^4 + x$

Answer

4x + 16 and 5x + 20

Answer

3x + 27 and $8x^3 + 72x^2$

Answer

$(x - y)^3$ and $x^3 - y^3$

Answer

$(a + b)^3$ and $a^3 + b^3$

Answer

$x^4 + 4$ and $2x^3 - 4x^2 + 4x$

Answer

$a^4 + a^2 + 1$ and $a^2 - a + 1$

Answer

$x^4 + x^2y^2 + y^4$ and $x^3 - y^3$

Answer

$1 + 4p + 4p^2 - 16p^4$ and $1 + 2p - 8p^3 - 16p^4$

Answer

$x^3 + x^2 + x + 1$ and $x^3 - x^2 + x - 1$

Answer

$y^4 + (2b^2 - a^2)y^2 + b^4$ and $y^3 - ay^2 + b^2y$

Answer

$\dfrac{x^4}{y^4} + \dfrac{y^4}{x^4} + 1$ and $\dfrac{x^3}{y^3} + \dfrac{y^3}{x^3}$

Hint

Answer

Question 6

Find the LCM.

$x^3 + 1$ , $x^4 - x^3 + x^2$ and $x^3 - x^2 + x$

Answer

$a^3 + 1$ , $a^4 + a^2 + 1$ and $a^2 + a + 1$

Answer

$x^2 - 3x + 2$ , $x^2 - 5x + 6$ and $x^2 - 8x + 12$

Answer

$2x^3 + 16$ , $x^2 + 4x + 4$ and $x^2 + 3x + 2$

Answer

$x^6 - 1$ , $x^3 - 1$ and $x^4 + x^2 + 1$

Answer

$x^6 - 16x^4$ , $x^5 + 6x^4 + 8x^3$ and $x^4 + 8x^3 + 16x^2$

Answer

$x^4 + 8x^2 + 144$ , $x^3 + x(x + 12) + 3x^2$ and $x^3 + 12x + 4x^2$

Answer

$x^4 - 8x^2 + 196$ , $x^3 + x(x + 14) + 5x^2$ and $2x^2 + 12x + 28$

Answer

$x^4 + 10x^2 + 169$ , $x^3 + 4x^2 + 13x$ and $x^3 + x(x + 13) + 3x^2$

Answer

$(y + 3)^2 - 9y - 27$ , $y^3 - 2y^2 - 15y$ and $y^5 - 13y^3 + 36y$

Answer

Question 7

Find the LCM.

$m^2 - 10m + 24 + 6n - 9n^2$ , $m^2 + 6mn + 9n^2 - 36$ and $m^2 + 3mn - 6m$

Answer

$x^4 - 8x^2 - 33 - 14y - y^2$ , $x^4 + 2x^2y - 9 + y^2$ and $x^3 + xy + 3x$

Answer

$a^4 + b^2(2a^2 - 1) + b^4$ , $a^3 - b(a + 1)(a - b) - b^3$ and $a^3 - b(a - 1)(a - b) - b^3$

Hint

Answer

Quiz yourself on this unit →

Practice every question in the book, and keep track of the ones you miss →

Where these facts come from

The cicadas (the opening, Back to the cicadas, the photo’s caption): Wikipedia, “Periodical cicadas” (13- and 17-year cycles; Broods XIII and XIX expected to emerge together in 2024, the first time since 1803, overlapping in central Illinois; the next such emergence in 2245; the hypotheses that prime cycles help avoid predators or keep broods from mixing; the nymphs’ years underground), read November 2026.
The photo of the cicada: Wikimedia Commons (CC BY-SA 3.0), credited under the picture.
How a computer finds an HCF (10.1, the fact): Wikipedia, “Polynomial greatest common divisor” (computed like the integer HCF by Euclid’s algorithm; factorizing is usually harder; computer algebra systems use HCFs to simplify fractions), read November 2026.
The 2,520 fact, the predator cycles and every answer: worked out here; every HCF and LCM checked with a computer algebra program (SymPy).
The exam’s question (For the exam, How the exam asks it): CDC, Specification Grid, Compulsory Mathematics, Grades 9–10 (Algebra 15 marks in three questions), and its Grade 9 model question 9a (the HCF of $x^3 + 27$ and $x^4 + 9x^2 + 81$ , 3 marks, understanding); our question is our own.

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