Highest factor, lowest multiple

Two numbers, a common language

Two questions that look schoolish, but they solve completely ordinary things:

Question 1: You have 12 apples and 18 pears. Into how many identical baskets can you split them so each has the same number of apples and the same number of pears — and as many baskets as possible? You are looking for the highest common factor (HCF).

Question 2: A bus runs every 4 minutes, a tram every 6 minutes. They just left together. How long until they leave together again? You are looking for the lowest common multiple (LCM).

A factor is a number that “fits into another with no remainder” (factors of 12: 1, 2, 3, 4, 6, 12). A multiple is a times-table result (multiples of 4: 4, 8, 12, 16…).

You look for HCF when you split something into equal parts. You look for LCM when something repeats and is meant to meet. And you already know one more use: HCF simplifies fractions, LCM is the best common denominator.

Ema is packing bags for the school fair: she has 12 gingerbreads and 18 lollipops. “I want as many identical bags as possible and nothing left over.” She tries 4 bags — the gingerbreads come out three each, but 18 lollipops into 4 does not go. Adam lists on paper: “Factors of twelve: 1, 2, 3, 4, 6, 12. Factors of eighteen: 1, 2, 3, 6, 9, 18. Common: 1, 2, 3 and 6. The highest is six.” Ema packs 6 bags: in each 2 gingerbreads and 3 lollipops. Nothing left, the bags are the same and there are as many as possible.

How to find HCF and LCM by listing

For numbers up to a hundred, honest listing is enough — no magic.

HCF — highest common factor:

1. List all the factors of the first number. For 12: 1, 2, 3, 4, 6, 12. (Look in pairs: 1×12, 2×6, 3×4 — then you forget none.)
2. List the factors of the second number. For 18: 1, 2, 3, 6, 9, 18.
3. Circle the common ones: 1, 2, 3, 6. Pick the highest: HCF = 6.

LCM — lowest common multiple:

1. List multiples of the bigger number (faster!). For 6: 6, 12, 18, 24…
2. At each one check whether it is also a multiple of the smaller number. Is 6 a multiple of four? No. Is 12? Yes!
3. The first hit is the LCM. LCM(4, 6) = 12.

Fast special cases:

• When one number divides the other (4 and 12): HCF is the smaller one (4), LCM the bigger one (12).
• When the numbers have no common factor except 1 (for example 4 and 9, they are called coprime): HCF = 1 and LCM is simply their product (36).

And the link with fractions: you simplify the fraction 12/18 by HCF → divided by six → 2/3. One step, no simplifying in stages.

Find HCF(24, 36)

Factors of 24 (in pairs 1×24, 2×12, 3×8, 4×6): 1, 2, 3, 4, 6, 8, 12, 24.

Factors of 36 (1×36, 2×18, 3×12, 4×9, 6×6): 1, 2, 3, 4, 6, 9, 12, 18, 36.

Common: 1, 2, 3, 4, 6, 12. Highest: HCF = 12.

Use at once: I simplify the fraction 24/36 by twelve to 2/3 in a single step. HCF is the strongest possible simplifying.

Bus and tram: find LCM(4, 6)

The bus goes every 4 minutes, the tram every 6. They left together at 14:00. When again?

Multiples of six (the bigger number): 6, 12, 18…

Test with four: 6 : 4 does not go cleanly. 12 : 4 = 3 cleanly → hit!

LCM = 12. They meet in 12 minutes, at 14:12. And then again at 14:24, 14:36 — every common multiple.

Meanwhile the bus left 3× (4, 8, 12), the tram 2× (6, 12). In the twelfth minute they met.

LCM as a common denominator: 1/6 + 3/8

From lesson ten you know the product of the denominators: 6 × 8 = 48. It works, but it is big. LCM gives a smaller one:

Multiples of 8: 8, 16, 24… Test with six: 8 no, 16 no, 24 yes → LCM = 24.

I expand: 1/6 = 4/24 and 3/8 = 9/24.

I add: 4/24 + 9/24 = 13/24.

With forty-eight it would come out 26/48 — the same value, but bigger numbers and extra simplifying. LCM = the smallest comfortable denominator.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Mulai with ten. When it goes well, add more.

Look for the highest common factor — list the factors of both numbers and pick the highest common one. The answer is a number.

Practise the questions

On paper

List honestly — pairs of factors and rows of multiples. Speed comes with practice.

1. Find HCF(16, 40) and HCF(15, 25). For both, list the factors in pairs.

2. Find LCM(3, 5), LCM(6, 9) and LCM(5, 10). Which of them was a special case and why?

3. Sofie has 20 stickers and 30 cards. How many identical bags can she make from them at most with nothing left — and what will be in one bag?

4. Jonáš swims every 3rd day, Tereza every 4th day. Today they met at the pool. In how many days will they meet there again? And add using LCM: 1/4 + 1/6.

Now you 💪

1. I can list all the factors of a number in pairs and skip none.
2. I can find LCM by listing multiples of the bigger number.
3. I know when to look for HCF (splitting) and when LCM (repeating, a common denominator).

What to take from this lesson

• HCF = the biggest number that divides both numbers. Look in the listed factors.
• LCM = the smallest number in the times table of both numbers. List multiples of the bigger one.
• HCF ≤ both numbers ≤ LCM — the fastest check that you did not mix them up.
• HCF simplifies fractions in one step, LCM is the smallest common denominator.
• Splitting into equal parts = HCF. Repeating events = LCM.

© 2026 Ing. Martin Polak / AlgoRhino · Ketentuan penggunaan konten